 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 Th111:
  |-_IPC (((p '&' q) => FALSUM) => FALSUM) => (((p => FALSUM) => FALSUM) '&'
  ((q => FALSUM) => FALSUM))
proof
A1: |-_IPC p '&' q => p by Th30;
A2: |-_IPC p '&' q => q by Th31;
A3: |-_IPC (p '&' q => p) => ((p => FALSUM) => ((p '&' q) => FALSUM)) by Th81;
A4: |-_IPC (p => FALSUM) => ((p '&' q) => FALSUM) by A1,A3,Th37;
    |-_IPC ((p => FALSUM) => ((p '&' q) => FALSUM)) =>
    ((((p '&' q) => FALSUM) => FALSUM) => ((p => FALSUM) => FALSUM))
      by Th81; then
A6: |-_IPC (((p '&' q) => FALSUM) => FALSUM) => ((p => FALSUM) => FALSUM)
      by A4,Th37;
    |-_IPC (p '&' q => q) => ((q => FALSUM) => ((p '&' q) => FALSUM)) by Th81;
      then
A8: |-_IPC (q => FALSUM) => ((p '&' q) => FALSUM) by A2,Th37;
    |-_IPC ((q => FALSUM) => ((p '&' q) => FALSUM)) =>
    ((((p '&' q) => FALSUM) => FALSUM) => ((q => FALSUM) => FALSUM))
      by Th81; then
A10: |-_IPC (((p '&' q) => FALSUM) => FALSUM) => ((q => FALSUM) => FALSUM)
       by A8,Th37;
    |-_IPC ((((p '&' q) => FALSUM) => FALSUM) => ((p => FALSUM) => FALSUM))
     => (((((p '&' q) => FALSUM) => FALSUM) => ((q => FALSUM) => FALSUM))
     => ((((p '&' q) => FALSUM) => FALSUM) =>
     (((p => FALSUM) => FALSUM) '&' ((q => FALSUM) => FALSUM))))
       by Th69,INTPRO_1:38; then
  |-_IPC ((((p '&' q) => FALSUM) => FALSUM) => ((q => FALSUM) => FALSUM))
     => ((((p '&' q) => FALSUM) => FALSUM) =>
     (((p => FALSUM) => FALSUM) '&' ((q => FALSUM) => FALSUM)))
       by A6,Th37;
  hence thesis by A10,Th37;
end;
