:: URYSOHN2 semantic presentation

theorem Th1: :: URYSOHN2:1
for b1 being Interval holds
( b1 <> {} implies ( ( ^^ b1 < b1 ^^ implies vol b1 = (b1 ^^ ) - (^^ b1) ) & ( b1 ^^ = ^^ b1 implies vol b1 = 0. ) ) )
proof end;

theorem Th2: :: URYSOHN2:2
for b1 being Subset of REAL
for b2 being Real holds
( b2 <> 0 implies (b2 " ) * (b2 * b1) = b1 )
proof end;

theorem Th3: :: URYSOHN2:3
for b1 being Real holds
( b1 <> 0 implies for b2 being Subset of REAL holds
( b2 = REAL implies b1 * b2 = b2 ) )
proof end;

theorem Th4: :: URYSOHN2:4
for b1 being Subset of REAL holds
( b1 <> {} implies 0 * b1 = {0} )
proof end;

theorem Th5: :: URYSOHN2:5
for b1 being Real holds b1 * {} = {}
proof end;

theorem Th6: :: URYSOHN2:6
for b1, b2 being R_eal holds
not ( b1 <= b2 & not ( b1 = -infty & b2 = -infty ) & not ( b1 = -infty & b2 in REAL ) & not ( b1 = -infty & b2 = +infty ) & not ( b1 in REAL & b2 in REAL ) & not ( b1 in REAL & b2 = +infty ) & not ( b1 = +infty & b2 = +infty ) )
proof end;

theorem Th7: :: URYSOHN2:7
for b1 being R_eal holds
[.b1,b1.] is Interval
proof end;

theorem Th8: :: URYSOHN2:8
for b1 being Interval holds
0 * b1 is Interval
proof end;

theorem Th9: :: URYSOHN2:9
for b1 being Interval
for b2 being Real holds
( b2 <> 0 & b1 is open_interval implies b2 * b1 is open_interval )
proof end;

theorem Th10: :: URYSOHN2:10
for b1 being Interval
for b2 being Real holds
( b2 <> 0 & b1 is closed_interval implies b2 * b1 is closed_interval )
proof end;

theorem Th11: :: URYSOHN2:11
for b1 being Interval
for b2 being Real holds
( 0 < b2 & b1 is right_open_interval implies b2 * b1 is right_open_interval )
proof end;

theorem Th12: :: URYSOHN2:12
for b1 being Interval
for b2 being Real holds
( b2 < 0 & b1 is right_open_interval implies b2 * b1 is left_open_interval )
proof end;

theorem Th13: :: URYSOHN2:13
for b1 being Interval
for b2 being Real holds
( 0 < b2 & b1 is left_open_interval implies b2 * b1 is left_open_interval )
proof end;

theorem Th14: :: URYSOHN2:14
for b1 being Interval
for b2 being Real holds
( b2 < 0 & b1 is left_open_interval implies b2 * b1 is right_open_interval )
proof end;

theorem Th15: :: URYSOHN2:15
for b1 being Interval holds
( b1 <> {} implies for b2 being Real holds
( 0 < b2 implies for b3 being Interval holds
( b3 = b2 * b1 & b1 = [.(^^ b1),(b1 ^^ ).] implies ( b3 = [.(^^ b3),(b3 ^^ ).] & ( for b4, b5 being Real holds
( b4 = ^^ b1 & b5 = b1 ^^ implies ( ^^ b3 = b2 * b4 & b3 ^^ = b2 * b5 ) ) ) ) ) ) )
proof end;

theorem Th16: :: URYSOHN2:16
for b1 being Interval holds
( b1 <> {} implies for b2 being Real holds
( 0 < b2 implies for b3 being Interval holds
( b3 = b2 * b1 & b1 = ].(^^ b1),(b1 ^^ ).] implies ( b3 = ].(^^ b3),(b3 ^^ ).] & ( for b4, b5 being Real holds
( b4 = ^^ b1 & b5 = b1 ^^ implies ( ^^ b3 = b2 * b4 & b3 ^^ = b2 * b5 ) ) ) ) ) ) )
proof end;

theorem Th17: :: URYSOHN2:17
for b1 being Interval holds
( b1 <> {} implies for b2 being Real holds
( 0 < b2 implies for b3 being Interval holds
( b3 = b2 * b1 & b1 = ].(^^ b1),(b1 ^^ ).[ implies ( b3 = ].(^^ b3),(b3 ^^ ).[ & ( for b4, b5 being Real holds
( b4 = ^^ b1 & b5 = b1 ^^ implies ( ^^ b3 = b2 * b4 & b3 ^^ = b2 * b5 ) ) ) ) ) ) )
proof end;

theorem Th18: :: URYSOHN2:18
for b1 being Interval holds
( b1 <> {} implies for b2 being Real holds
( 0 < b2 implies for b3 being Interval holds
( b3 = b2 * b1 & b1 = [.(^^ b1),(b1 ^^ ).[ implies ( b3 = [.(^^ b3),(b3 ^^ ).[ & ( for b4, b5 being Real holds
( b4 = ^^ b1 & b5 = b1 ^^ implies ( ^^ b3 = b2 * b4 & b3 ^^ = b2 * b5 ) ) ) ) ) ) )
proof end;

theorem Th19: :: URYSOHN2:19
for b1 being Interval
for b2 being Real holds
b2 * b1 is Interval
proof end;

registration
let c1 be Interval;
let c2 be Real;
cluster a2 * a1 -> interval ;
correctness
coherence
c2 * c1 is interval
;
by Th19;
end;

theorem Th20: :: URYSOHN2:20
for b1 being Interval
for b2 being Real holds
( 0 <= b2 implies for b3 being Real holds
( b3 = vol b1 implies b2 * b3 = vol (b2 * b1) ) )
proof end;

theorem Th21: :: URYSOHN2:21
canceled;

theorem Th22: :: URYSOHN2:22
canceled;

theorem Th23: :: URYSOHN2:23
for b1 being Real holds
not ( 0 < b1 & ( for b2 being Nat holds not 1 < (2 |^ b2) * b1 ) )
proof end;

theorem Th24: :: URYSOHN2:24
for b1, b2 being Real holds
not ( 0 <= b1 & 1 < b2 - b1 & ( for b3 being Nat holds
not ( b1 < b3 & b3 < b2 ) ) )
proof end;

theorem Th25: :: URYSOHN2:25
canceled;

theorem Th26: :: URYSOHN2:26
canceled;

theorem Th27: :: URYSOHN2:27
for b1 being Nat holds dyadic b1 c= DYADIC
proof end;

theorem Th28: :: URYSOHN2:28
for b1, b2 being Real holds
not ( b1 < b2 & 0 <= b1 & b2 <= 1 & ( for b3 being Real holds
not ( b3 in DYADIC & b1 < b3 & b3 < b2 ) ) )
proof end;

theorem Th29: :: URYSOHN2:29
for b1, b2 being Real holds
not ( b1 < b2 & ( for b3 being Real holds
not ( b3 in DOM & b1 < b3 & b3 < b2 ) ) )
proof end;

theorem Th30: :: URYSOHN2:30
for b1 being non empty Subset of ExtREAL
for b2, b3 being R_eal holds
( b1 c= [.b2,b3.] implies ( b2 <= inf b1 & sup b1 <= b3 ) )
proof end;

theorem Th31: :: URYSOHN2:31
( 0 in DYADIC & 1 in DYADIC )
proof end;

theorem Th32: :: URYSOHN2:32
for b1, b2 being R_eal holds
( b1 = 0 & b2 = 1 implies DYADIC c= [.b1,b2.] )
proof end;

theorem Th33: :: URYSOHN2:33
for b1, b2 being Nat holds
( b1 <= b2 implies dyadic b1 c= dyadic b2 )
proof end;

theorem Th34: :: URYSOHN2:34
for b1, b2, b3, b4 being Real holds
not ( b1 < b3 & b3 < b2 & b1 < b4 & b4 < b2 & not abs (b4 - b3) < b2 - b1 )
proof end;

theorem Th35: :: URYSOHN2:35
for b1 being Real holds
( 0 < b1 implies for b2 being Real holds
not ( 0 < b2 & b2 <= 1 & ( for b3, b4 being Real holds
not ( b3 in DYADIC \/ (right_open_halfline 1) & b4 in DYADIC \/ (right_open_halfline 1) & 0 < b3 & b3 < b2 & b2 < b4 & b4 - b3 < b1 ) ) ) )
proof end;