reserve T, X, Y for Subset of MC-wff;
reserve p, q, r, s for Element of MC-wff;
reserve T, X, Y for Subset of MC-wff;
reserve p, q, r for Element of MC-wff;

theorem Th69:
  p => (q => p) in CnCPC (X) & (p => (q => r)) => ((p => q) => (p
=> r)) in CnCPC (X) & p '&' q => p in CnCPC (X) & p '&' q => q in CnCPC (X) & p
=> (q => (p '&' q)) in CnCPC (X) & p => (p 'or' q) in CnCPC (X) & q => (p 'or'
  q) in CnCPC (X) & (p => r) => ((q => r) => ((p 'or' q) => r)) in CnCPC (X) &
  FALSUM => p in CnCPC (X) & p 'or' (p => FALSUM) in CnCPC (X)
proof
A1: CnIPC(X) c= CnCPC(X) by Th68;
  p => (q => p) in CnIPC (X) by Th1;
  hence p => (q => p) in CnCPC (X) by A1;
A2: CnIPC(X) c= CnCPC(X) by Th68;
  (p => (q => r)) => ((p => q) => (p => r)) in CnIPC (X) by Th2;
  hence (p => (q => r)) => ((p => q) => (p => r)) in CnCPC (X) by A2;
A3: CnIPC(X) c= CnCPC(X) by Th68;
  p '&' q => p in CnIPC (X) by Th3;
  hence p '&' q => p in CnCPC (X) by A3;
A4: CnIPC(X) c= CnCPC(X) by Th68;
  p '&' q => q in CnIPC (X) by Th4;
  hence p '&' q => q in CnCPC (X) by A4;
A5: CnIPC(X) c= CnCPC(X) by Th68;
  p => (q => (p '&' q)) in CnIPC (X) by Th5;
  hence p => (q => (p '&' q)) in CnCPC (X) by A5;
A6: CnIPC(X) c= CnCPC(X) by Th68;
  p => (p 'or' q) in CnIPC (X) by Th6;
  hence p => (p 'or' q) in CnCPC (X) by A6;
A7: CnIPC(X) c= CnCPC(X) by Th68;
  q => (p 'or' q) in CnIPC (X) by Th7;
  hence q => (p 'or' q) in CnCPC (X) by A7;
A8: CnIPC(X) c= CnCPC(X) by Th68;
  (p => r) => ((q => r) => ((p 'or' q) => r)) in CnIPC (X) by Th8;
  hence (p => r) => ((q => r) => ((p 'or' q) => r)) in CnCPC (X) by A8;
A9: CnIPC(X) c= CnCPC(X) by Th68;
  FALSUM => p in CnIPC (X) by Th9;
  hence FALSUM => p in CnCPC (X) by A9;
  T is CPC_theory & X c= T implies p 'or' (p => FALSUM) in T;
  hence p 'or' (p => FALSUM) in CnCPC (X) by Def20;
end;
