 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 Th105:
  |-_IPC ((p => FALSUM) => q) => ((((p => FALSUM) => FALSUM) => FALSUM) => q)
proof
    ((p => FALSUM) => FALSUM) => FALSUM in
     {((p => FALSUM) => FALSUM) => FALSUM,(p => FALSUM) => q}
       by TARSKI:def 2; then
A1: {((p => FALSUM) => FALSUM) => FALSUM,(p => FALSUM) => q}
    |-_IPC ((p => FALSUM) => FALSUM) => FALSUM by Th67;
    (p => FALSUM) =>q in {((p => FALSUM) => FALSUM) => FALSUM,
     (p => FALSUM) => q} by TARSKI:def 2; then
A2: {((p => FALSUM) => FALSUM) => FALSUM,(p => FALSUM) => q}
    |-_IPC (p => FALSUM) => q by Th67;
A03: {}(MC-wff) c= {((p => FALSUM) => FALSUM) => FALSUM,(p => FALSUM) => q};
    |-_IPC (((p => FALSUM) => FALSUM) => FALSUM) => (p => FALSUM) by Th103;
    then
A3: {((p => FALSUM) => FALSUM) => FALSUM,(p => FALSUM) => q}
    |-_IPC (((p => FALSUM) => FALSUM) => FALSUM) => (p => FALSUM)
      by A03,Th66;
    {((p => FALSUM) => FALSUM) => FALSUM,(p => FALSUM) => q}
    |-_IPC p => FALSUM by A1,A3,Th27; then
   {((p => FALSUM) => FALSUM) => FALSUM,(p => FALSUM) => q}
    |-_IPC q by A2,Th27; then
  {(p => FALSUM) => q}
    |-_IPC (((p => FALSUM) => FALSUM) => FALSUM) => q by Th55;
  hence thesis by Th54;
end;
