 reserve i,j,n,k,l for Nat;
 reserve T,S,X,Y,Z for Subset of MC-wff;
 reserve p,q,r,t,F,H,G for Element of MC-wff;
 reserve s,U,V for MC-formula;
reserve f,g for FinSequence of [:MC-wff,Proof_Step_Kinds_IPC:];
 reserve X,T for Subset of MC-wff;
 reserve F,G,H,p,q,r,t for Element of MC-wff;
 reserve s,h for MC-formula;
 reserve f for FinSequence of [:MC-wff,Proof_Step_Kinds_IPC:];
 reserve i,j for Element of NAT;
 reserve F1,F2,F3,F4,F5,F6,F7,F8,F9,F10,G for MC-formula;
 reserve x1,x2,x3,x4,x5,x6,x7,x8,x9,x10,x for Element of MC-wff;
reserve x1,x2,x3,x4,x5,x6,x7,x8,x9,x10 for object;

theorem Th82:
  |-_IPC ((p => FALSUM)'or' (q => FALSUM)) => ((p '&' q)=> FALSUM)
proof
A1: (((p => FALSUM) =>((p '&' q) => FALSUM)) '&'
  ((q => FALSUM) =>((p '&' q) => FALSUM))) =>
  (((p => FALSUM) 'or' (q => FALSUM)) => ((p '&' q) => FALSUM))
  in IPC-Taut by Th70,Th77;
A2: (p '&' q) => p in IPC-Taut by INTPRO_1:def 14;
  ((p '&' q) => p) => ((p => FALSUM) => ((p '&' q) => FALSUM)) in IPC-Taut
    by Th70,Th81; then
A3: (p => FALSUM) => ((p '&' q) => FALSUM) in IPC-Taut
    by A2,INTPRO_1:def 14;
A4: (p '&' q) => q in IPC-Taut by INTPRO_1:def 14;
  ((p '&' q) => q) => ((q => FALSUM) => ((p '&' q) => FALSUM))
  in IPC-Taut by Th70,Th81; then
   (q => FALSUM) => ((p '&' q) => FALSUM) in IPC-Taut
    by A4,INTPRO_1:def 14; then
  ((p => FALSUM) =>((p '&' q) => FALSUM)) '&'
  ((q => FALSUM) =>((p '&' q) => FALSUM)) in IPC-Taut by A3,INTPRO_1:34;
  then ((p => FALSUM) 'or' (q => FALSUM)) => ((p '&' q)=> FALSUM)
    in IPC-Taut by A1,INTPRO_1:def 14;
  hence thesis by Th69;
end;
