:: TOPMETR3 semantic presentation

theorem Th1: :: TOPMETR3:1
for b1 being non empty Subset of REAL
for b2 being real number holds
( ( for b3 being real number holds
( b3 in b1 implies b3 <= b2 ) ) implies upper_bound b1 <= b2 )
proof end;

theorem Th2: :: TOPMETR3:2
for b1 being non empty MetrSpace
for b2 being sequence of b1
for b3 being Subset of (TopSpaceMetr b1) holds
( b2 is convergent & ( for b4 being Nat holds b2 . b4 in b3 ) & b3 is closed implies lim b2 in b3 )
proof end;

theorem Th3: :: TOPMETR3:3
for b1, b2 being non empty MetrSpace
for b3 being Function of (TopSpaceMetr b1),(TopSpaceMetr b2)
for b4 being sequence of b1 holds
b3 * b4 is sequence of b2
proof end;

theorem Th4: :: TOPMETR3:4
for b1, b2 being non empty MetrSpace
for b3 being Function of (TopSpaceMetr b1),(TopSpaceMetr b2)
for b4 being sequence of b1
for b5 being sequence of b2 holds
( b4 is convergent & b5 = b3 * b4 & b3 is continuous implies b5 is convergent )
proof end;

theorem Th5: :: TOPMETR3:5
canceled;

theorem Th6: :: TOPMETR3:6
for b1 being Real_Sequence
for b2 being sequence of RealSpace holds
( b1 = b2 implies ( ( b1 is convergent implies b2 is convergent ) & ( b2 is convergent implies b1 is convergent ) & ( b1 is convergent implies lim b1 = lim b2 ) ) )
proof end;

theorem Th7: :: TOPMETR3:7
for b1, b2 being real number
for b3 being Real_Sequence holds
( rng b3 c= [.b1,b2.] implies b3 is sequence of (Closed-Interval-MSpace b1,b2) )
proof end;

theorem Th8: :: TOPMETR3:8
for b1, b2 being real number
for b3 being sequence of (Closed-Interval-MSpace b1,b2) holds
( b1 <= b2 implies b3 is sequence of RealSpace )
proof end;

theorem Th9: :: TOPMETR3:9
for b1, b2 being real number
for b3 being sequence of (Closed-Interval-MSpace b1,b2)
for b4 being sequence of RealSpace holds
( b4 = b3 & b1 <= b2 implies ( ( b4 is convergent implies b3 is convergent ) & ( b3 is convergent implies b4 is convergent ) & ( b4 is convergent implies lim b4 = lim b3 ) ) )
proof end;

theorem Th10: :: TOPMETR3:10
for b1, b2 being real number
for b3 being Real_Sequence
for b4 being sequence of (Closed-Interval-MSpace b1,b2) holds
( b4 = b3 & b1 <= b2 & b3 is convergent implies ( b4 is convergent & lim b3 = lim b4 ) )
proof end;

theorem Th11: :: TOPMETR3:11
for b1, b2 being real number
for b3 being Real_Sequence
for b4 being sequence of (Closed-Interval-MSpace b1,b2) holds
( b4 = b3 & b1 <= b2 & b3 is non-decreasing implies b4 is convergent )
proof end;

theorem Th12: :: TOPMETR3:12
for b1, b2 being real number
for b3 being Real_Sequence
for b4 being sequence of (Closed-Interval-MSpace b1,b2) holds
( b4 = b3 & b1 <= b2 & b3 is non-increasing implies b4 is convergent )
proof end;

theorem Th13: :: TOPMETR3:13
canceled;

theorem Th14: :: TOPMETR3:14
canceled;

theorem Th15: :: TOPMETR3:15
for b1 being non empty Subset of REAL holds
not ( b1 is bounded_above & ( for b2 being Real_Sequence holds
not ( b2 is non-decreasing & b2 is convergent & rng b2 c= b1 & lim b2 = upper_bound b1 ) ) )
proof end;

theorem Th16: :: TOPMETR3:16
for b1 being non empty Subset of REAL holds
not ( b1 is bounded_below & ( for b2 being Real_Sequence holds
not ( b2 is non-increasing & b2 is convergent & rng b2 c= b1 & lim b2 = lower_bound b1 ) ) )
proof end;

theorem Th17: :: TOPMETR3:17
for b1 being non empty MetrSpace
for b2 being Function of I[01] ,(TopSpaceMetr b1)
for b3, b4 being Subset of (TopSpaceMetr b1)
for b5, b6 being Real holds
not ( 0 <= b5 & b6 <= 1 & b5 <= b6 & b2 . b5 in b3 & b2 . b6 in b4 & b3 is closed & b4 is closed & b2 is continuous & b3 \/ b4 = the carrier of b1 & ( for b7 being Real holds
not ( b5 <= b7 & b7 <= b6 & b2 . b7 in b3 /\ b4 ) ) )
proof end;

theorem Th18: :: TOPMETR3:18
for b1 being Nat
for b2, b3 being Point of (TOP-REAL b1)
for b4, b5 being non empty Subset of (TOP-REAL b1) holds
( b4 is_an_arc_of b2,b3 & b5 is_an_arc_of b3,b2 & b5 c= b4 implies b5 = b4 )
proof end;

theorem Th19: :: TOPMETR3:19
for b1, b2 being non empty compact Subset of (TOP-REAL 2) holds
not ( b1 is_simple_closed_curve & b2 is_an_arc_of W-min b1, E-max b1 & b2 c= b1 & not b2 = Upper_Arc b1 & not b2 = Lower_Arc b1 )
proof end;