 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 Th108:
  |-_IPC (((p => q) => FALSUM) => FALSUM) => (((p => FALSUM) => FALSUM) =>
  ((q => FALSUM) => FALSUM))
proof
A01: p in {p,q => FALSUM,(p => FALSUM) => FALSUM,
     ((p => q) => FALSUM) => FALSUM} by ENUMSET1:def 2;
A1: {p,q => FALSUM,(p => FALSUM) => FALSUM,((p => q) => FALSUM) => FALSUM}
    |-_IPC p by A01,Th67;
    q => FALSUM in
     {p,q => FALSUM,(p => FALSUM) => FALSUM,((p => q) => FALSUM) => FALSUM}
       by ENUMSET1:def 2; then
A2: {p,q => FALSUM,(p => FALSUM) => FALSUM,((p => q) => FALSUM) => FALSUM}
    |-_IPC q => FALSUM by Th67;
    ((p => q) => FALSUM) => FALSUM in
     {p,q => FALSUM,(p => FALSUM) => FALSUM,((p => q) => FALSUM) => FALSUM}
       by ENUMSET1:def 2; then
A3: {p,q => FALSUM,(p => FALSUM) => FALSUM,((p => q) => FALSUM) => FALSUM}
    |-_IPC ((p => q) => FALSUM) => FALSUM by Th67;
    {p,q => FALSUM,(p => FALSUM) => FALSUM,((p => q) => FALSUM) => FALSUM}
     |-_IPC p => ((q => FALSUM) => (p '&' (q => FALSUM))) by Th22; then
    {p,q => FALSUM,(p => FALSUM) => FALSUM,
     ((p => q) => FALSUM) => FALSUM}
     |-_IPC ((q => FALSUM) => (p '&' (q => FALSUM))) by A1,Th27; then
A4: {p,q => FALSUM,(p => FALSUM) => FALSUM,((p => q) => FALSUM) => FALSUM}
    |-_IPC p '&' (q => FALSUM) by A2,Th27;
A06: |-_IPC (p '&' (q => FALSUM)) => ((p => q) => FALSUM) by Th107;
A07: {}(MC-wff) c= {p,q => FALSUM,(p => FALSUM) => FALSUM,
     ((p => q) => FALSUM) => FALSUM};
     |-_IPC ((p '&' (q => FALSUM)) => ((p => q) => FALSUM))
     => ((((p => q) => FALSUM) => FALSUM) =>
     ((p '&' (q => FALSUM)) => FALSUM)) by Th81; then
     |-_IPC (((p => q) => FALSUM) => FALSUM) =>
     ((p '&' (q => FALSUM)) => FALSUM) by A06,Th37; then
     {p,q => FALSUM,(p => FALSUM) => FALSUM,((p => q) => FALSUM) => FALSUM}
    |-_IPC (((p => q) => FALSUM) => FALSUM) =>
    ((p '&' (q => FALSUM)) => FALSUM) by A07,Th66; then
    {p,q => FALSUM,(p => FALSUM) => FALSUM,((p => q) => FALSUM) => FALSUM}
    |-_IPC (p '&' (q => FALSUM)) => FALSUM by A3,Th27; then
    {p,q => FALSUM,(p => FALSUM) => FALSUM,
    ((p => q) => FALSUM) => FALSUM} |-_IPC FALSUM by A4,Th27; then
A8: {q => FALSUM,(p => FALSUM) => FALSUM,((p => q) => FALSUM) => FALSUM}
    |-_IPC p => FALSUM by Th57;
    (p => FALSUM) => FALSUM in
     {q => FALSUM,(p => FALSUM) => FALSUM,((p => q) => FALSUM) => FALSUM}
       by ENUMSET1:def 1; then
    {q => FALSUM,(p => FALSUM) => FALSUM,((p => q) => FALSUM) => FALSUM}
      |-_IPC (p => FALSUM) => FALSUM by Th67; then
    {q => FALSUM,(p => FALSUM) => FALSUM,((p => q) => FALSUM) => FALSUM}
      |-_IPC FALSUM by A8,Th27; then
    {(p => FALSUM) => FALSUM,((p => q) => FALSUM) => FALSUM}
      |-_IPC (q => FALSUM) => FALSUM by Th56; then
    {((p => q) => FALSUM) => FALSUM}
      |-_IPC ((p => FALSUM) => FALSUM) => ((q => FALSUM) => FALSUM) by Th55;
    hence thesis by Th54;
end;
