theorem Th7:
  X is Inconsistent implies ex Y st Y c= X & Y is finite & Y is Inconsistent
proof
  assume X is Inconsistent;
  then consider p such that
A1: X |- p and
A2: X |- 'not' p;
  consider f1 such that
A3: rng f1 c= X and
A4: |- f1^<*p*> by A1;
  consider f2 such that
A5: rng f2 c= X and
A6: |- f2^<*'not' p*> by A2;
  reconsider Y = rng(f1^f2) as Subset of CQC-WFF(Al);
  take Y;
  Y = rng f1 \/ rng f2 by FINSEQ_1:31;
  hence Y c= X by A3,A5,XBOOLE_1:8;
  |- f1^f2^<*'not' p*> by A6,CALCUL_2:20;
  then
A7: Y |- 'not' p;
  |- f1^f2^<*p*> by A4,Th5;
  then Y |- p;
  hence thesis by A7;
end;
