 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 Th109:
  |-_IPC (((p => FALSUM) => FALSUM) => ((q => FALSUM) => FALSUM)) =>
  (((p => q) => FALSUM) => FALSUM)
proof
    (p => q) => FALSUM in
     {(p => q) => FALSUM,((p => FALSUM) => FALSUM) =>
     ((q => FALSUM) => FALSUM)} by TARSKI:def 2; then
A1: {(p => q) => FALSUM,((p => FALSUM) => FALSUM) => ((q => FALSUM) => FALSUM)}
    |-_IPC (p => q) => FALSUM by Th67;
    ((p => FALSUM) => FALSUM) => ((q => FALSUM) => FALSUM) in
     {(p => q) => FALSUM,((p => FALSUM) => FALSUM) =>
     ((q => FALSUM) => FALSUM)} by TARSKI:def 2; then
A2: {(p => q) => FALSUM,((p => FALSUM) => FALSUM) => ((q => FALSUM) => FALSUM)}
    |-_IPC ((p => FALSUM) => FALSUM) => ((q => FALSUM) => FALSUM) by Th67;
A03: |-_IPC ((p => FALSUM) 'or' q) => (p => q) by Th80;
A04: {}(MC-wff) c= {(p => q) => FALSUM,
     ((p => FALSUM) => FALSUM) => ((q => FALSUM) => FALSUM)};
     |-_IPC (((p => FALSUM) 'or' q) => (p => q)) =>
     (((p => q) => FALSUM) => (((p => FALSUM) 'or' q) => FALSUM)) by Th81; then
     |-_IPC ((p => q) => FALSUM) => (((p => FALSUM) 'or' q) => FALSUM)
       by A03,Th37; then
     {(p => q) => FALSUM,((p => FALSUM) => FALSUM) => ((q => FALSUM)
       => FALSUM)}
    |-_IPC ((p => q) => FALSUM) => (((p => FALSUM) 'or' q) => FALSUM)
      by A04,Th66; then
A4: {(p => q) => FALSUM,((p => FALSUM) => FALSUM) => ((q => FALSUM) => FALSUM)}
    |-_IPC ((p => FALSUM) 'or' q) => FALSUM by A1,Th27;
A07: |-_IPC (((p => FALSUM) 'or' q) => FALSUM) =>
     (((p => FALSUM) => FALSUM) '&' (q => FALSUM)) by Th97;
    {(p => q) => FALSUM,((p => FALSUM) => FALSUM) => ((q => FALSUM) => FALSUM)}
    |-_IPC (((p => FALSUM) 'or' q) => FALSUM) =>
    (((p => FALSUM)=> FALSUM) '&' (q => FALSUM)) by A04,A07,Th66; then
A6: {(p => q) => FALSUM,((p => FALSUM) => FALSUM) => ((q => FALSUM) => FALSUM)}
    |-_IPC ((p => FALSUM) => FALSUM) '&' (q => FALSUM) by A4,Th27;
    {(p => q) => FALSUM,((p => FALSUM) => FALSUM) =>
     ((q => FALSUM) => FALSUM)} |-_IPC (((p => FALSUM)=> FALSUM) '&'
     (q => FALSUM)) => ((p => FALSUM) => FALSUM) by Th20; then
    {(p => q) => FALSUM,((p => FALSUM) => FALSUM) => ((q => FALSUM) => FALSUM)}
    |-_IPC (p => FALSUM) => FALSUM by A6,Th27; then
A8: {(p => q) => FALSUM,((p => FALSUM) => FALSUM) => ((q => FALSUM) => FALSUM)}
    |-_IPC (q => FALSUM)=> FALSUM by A2,Th27;
    {(p => q) => FALSUM,((p => FALSUM) => FALSUM) =>
     ((q => FALSUM) => FALSUM)} |-_IPC (((p => FALSUM) => FALSUM) '&'
     (q => FALSUM)) => (q => FALSUM) by Th21; then
    {(p => q) => FALSUM,((p => FALSUM) => FALSUM) => ((q => FALSUM) => FALSUM)}
    |-_IPC q => FALSUM by A6,Th27; then
    {(p => q) => FALSUM,((p => FALSUM) => FALSUM) =>
     ((q => FALSUM) => FALSUM)} |-_IPC FALSUM by A8,Th27; then
    {((p => FALSUM)=> FALSUM) => ((q => FALSUM) => FALSUM)}
     |-_IPC ((p => q) => FALSUM) => FALSUM by Th55;
    hence thesis by Th54;
end;
