:: FINSET_1 semantic presentation
:: deftheorem Def1 defines finite FINSET_1:def 1 :
Lemma25:
for x being set holds {x} is finite
Lemma33:
for A, B being set st A is finite & B is finite holds
A \/ B is finite
registration
let x1 be
set ,
x2 be
set ,
x3 be
set ,
x4 be
set ,
x5 be
set ;
cluster {a1,a2,a3,a4,a5} -> finite ;
coherence
{x1,x2,x3,x4,x5} is finite
end;
registration
let x1 be
set ,
x2 be
set ,
x3 be
set ,
x4 be
set ,
x5 be
set ,
x6 be
set ;
cluster {a1,a2,a3,a4,a5,a6} -> finite ;
coherence
{x1,x2,x3,x4,x5,x6} is finite
end;
registration
let x1 be
set ,
x2 be
set ,
x3 be
set ,
x4 be
set ,
x5 be
set ,
x6 be
set ,
x7 be
set ;
cluster {a1,a2,a3,a4,a5,a6,a7} -> finite ;
coherence
{x1,x2,x3,x4,x5,x6,x7} is finite
end;
registration
let x1 be
set ,
x2 be
set ,
x3 be
set ,
x4 be
set ,
x5 be
set ,
x6 be
set ,
x7 be
set ,
x8 be
set ;
cluster {a1,a2,a3,a4,a5,a6,a7,a8} -> finite ;
coherence
{x1,x2,x3,x4,x5,x6,x7,x8} is finite
end;
theorem Th1: :: FINSET_1:1
canceled;
theorem Th2: :: FINSET_1:2
canceled;
theorem Th3: :: FINSET_1:3
canceled;
theorem Th4: :: FINSET_1:4
canceled;
theorem Th5: :: FINSET_1:5
canceled;
theorem Th6: :: FINSET_1:6
canceled;
theorem Th7: :: FINSET_1:7
canceled;
theorem Th8: :: FINSET_1:8
canceled;
theorem Th9: :: FINSET_1:9
canceled;
theorem Th10: :: FINSET_1:10
canceled;
theorem Th11: :: FINSET_1:11
canceled;
theorem Th12: :: FINSET_1:12
canceled;
theorem Th13: :: FINSET_1:13
theorem Th14: :: FINSET_1:14
theorem Th15: :: FINSET_1:15
theorem Th16: :: FINSET_1:16
theorem Th17: :: FINSET_1:17
theorem Th18: :: FINSET_1:18
Lemma106:
for A being set st A is finite & ( for X being set st X in A holds
X is finite ) holds
union A is finite
theorem Th19: :: FINSET_1:19
theorem Th20: :: FINSET_1:20
theorem Th21: :: FINSET_1:21
theorem Th22: :: FINSET_1:22
theorem Th23: :: FINSET_1:23
theorem Th24: :: FINSET_1:24
theorem Th25: :: FINSET_1:25
theorem Th26: :: FINSET_1:26
theorem Th27: :: FINSET_1:27
theorem Th28: :: FINSET_1:28
theorem Th29: :: FINSET_1:29
theorem Th30: :: FINSET_1:30
theorem Th31: :: FINSET_1:31
:: deftheorem Def2 defines finite-yielding FINSET_1:def 2 :