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

theorem Th82:
  p => (q => p) in CnS4 (X) & (p => (q => r)) => ((p => q) => (p
=> r)) in CnS4 (X) & p '&' q => p in CnS4 (X) & p '&' q => q in CnS4 (X) & p =>
(q => (p '&' q)) in CnS4 (X) & p => (p 'or' q) in CnS4 (X) & q => (p 'or' q) in
CnS4 (X) & (p => r) => ((q => r) => ((p 'or' q) => r)) in CnS4 (X) & FALSUM =>
  p in CnS4 (X) & p 'or' (p => FALSUM) in CnS4 (X)
proof
A1: CnIPC (X) c= CnS4 (X) by Th81;
  p => (q => p) in CnIPC (X) by Th1;
  hence p => (q => p) in CnS4 (X) by A1;
A2: CnIPC (X) c= CnS4 (X) by Th81;
  (p => (q => r)) => ((p => q) => (p => r)) in CnIPC (X) by Th2;
  hence (p => (q => r)) => ((p => q) => (p => r)) in CnS4 (X) by A2;
A3: CnIPC (X) c= CnS4 (X) by Th81;
  p '&' q => p in CnIPC (X) by Th3;
  hence p '&' q => p in CnS4 (X) by A3;
A4: CnIPC (X) c= CnS4 (X) by Th81;
  p '&' q => q in CnIPC (X) by Th4;
  hence p '&' q => q in CnS4 (X) by A4;
A5: CnIPC (X) c= CnS4 (X) by Th81;
  p => (q => (p '&' q)) in CnIPC (X) by Th5;
  hence p => (q => (p '&' q)) in CnS4 (X) by A5;
A6: CnIPC (X) c= CnS4 (X) by Th81;
  p => (p 'or' q) in CnIPC (X) by Th6;
  hence p => (p 'or' q) in CnS4 (X) by A6;
A7: CnIPC (X) c= CnS4 (X) by Th81;
  q => (p 'or' q) in CnIPC (X) by Th7;
  hence q => (p 'or' q) in CnS4 (X) by A7;
A8: CnIPC (X) c= CnS4 (X) by Th81;
  (p => r) => ((q => r) => ((p 'or' q) => r)) in CnIPC (X) by Th8;
  hence (p => r) => ((q => r) => ((p 'or' q) => r)) in CnS4 (X) by A8;
A9: CnIPC (X) c= CnS4 (X) by Th81;
  FALSUM => p in CnIPC (X) by Th9;
  hence FALSUM => p in CnS4 (X) by A9;
  T is S4_theory & X c= T implies p 'or' (p => FALSUM) in T;
  hence p 'or' (p => FALSUM) in CnS4 (X) by Def23;
end;
