 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 Th112:
  |-_IPC (((p => FALSUM) => FALSUM) '&' ((q => FALSUM) => FALSUM)) =>
  (((p '&' q) => FALSUM) => FALSUM)
proof
    p in {p,q,(p '&' q) => FALSUM,
     ((p => FALSUM) => FALSUM) '&' ((q => FALSUM) => FALSUM)}
       by ENUMSET1:def 2; then
A1: {p,q,(p '&' q) => FALSUM,
    ((p => FALSUM) => FALSUM) '&' ((q => FALSUM) => FALSUM)} |-_IPC p
      by Th67;
    q in {p,q,(p '&' q) => FALSUM,
     ((p => FALSUM) => FALSUM) '&' ((q => FALSUM) => FALSUM)}
       by ENUMSET1:def 2; then
A2: {p,q,(p '&' q) => FALSUM,
    ((p => FALSUM) => FALSUM) '&' ((q => FALSUM) => FALSUM)} |-_IPC q
      by Th67;
    (p '&' q) => FALSUM in {p,q,(p '&' q) => FALSUM,
     ((p => FALSUM) => FALSUM) '&' ((q => FALSUM) => FALSUM)}
       by ENUMSET1:def 2; then
A3: {p,q,(p '&' q) => FALSUM,
    ((p => FALSUM) => FALSUM) '&' ((q => FALSUM) => FALSUM)}
    |-_IPC (p '&' q)=> FALSUM by Th67;
    {p,q,(p '&' q) => FALSUM,
     ((p => FALSUM) => FALSUM) '&' ((q => FALSUM) => FALSUM)}
     |-_IPC p => (q => (p '&' q)) by Th22; then
    {p,q,(p '&' q) => FALSUM,
     ((p => FALSUM) => FALSUM) '&' ((q => FALSUM) => FALSUM)}
     |-_IPC (q => (p '&' q)) by A1,Th27; then
    {p,q,(p '&' q) => FALSUM,
    ((p => FALSUM) => FALSUM) '&' ((q => FALSUM) => FALSUM)}
    |-_IPC p '&' q by A2,Th27; then
    {p,q,(p '&' q) => FALSUM,
    ((p => FALSUM) => FALSUM) '&' ((q => FALSUM) => FALSUM)}
    |-_IPC FALSUM by A3,Th27; then
A6: {q,(p '&' q) => FALSUM,
    ((p => FALSUM) => FALSUM) '&' ((q => FALSUM) => FALSUM)}
    |-_IPC p => FALSUM by Th57;
    ((p => FALSUM) => FALSUM) '&' ((q => FALSUM) => FALSUM) in {q,
     (p '&' q) => FALSUM,((p => FALSUM) => FALSUM) '&'
     ((q => FALSUM) => FALSUM)} by ENUMSET1:def 1; then
B1: {q,(p '&' q) => FALSUM,
    ((p => FALSUM) => FALSUM) '&' ((q => FALSUM) => FALSUM)}
    |-_IPC ((p => FALSUM) => FALSUM) '&' ((q => FALSUM) => FALSUM) by Th67;
    {q,(p '&' q) => FALSUM,
     ((p => FALSUM) => FALSUM) '&' ((q => FALSUM) => FALSUM)}
     |-_IPC (((p => FALSUM) => FALSUM) '&' ((q => FALSUM) => FALSUM))
     => ((p => FALSUM) => FALSUM) by Th20; then
    {q,(p '&' q) => FALSUM,
    ((p => FALSUM) => FALSUM) '&' ((q => FALSUM) => FALSUM)}
    |-_IPC (p => FALSUM) => FALSUM by B1,Th27; then
B3: {q,(p '&' q) => FALSUM,
    ((p => FALSUM) => FALSUM) '&' ((q => FALSUM) => FALSUM)}
    |-_IPC FALSUM by A6,Th27;
B4: {(p '&' q) => FALSUM,
    ((p => FALSUM) => FALSUM) '&' ((q => FALSUM) => FALSUM)}
    |-_IPC q => FALSUM by B3,Th56;
    ((p => FALSUM) => FALSUM) '&' ((q => FALSUM) => FALSUM) in {
     (p '&' q) => FALSUM,((p => FALSUM) => FALSUM) '&'
     ((q => FALSUM) => FALSUM)} by TARSKI:def 2; then
C1: {(p '&' q) => FALSUM,
    ((p => FALSUM) => FALSUM) '&' ((q => FALSUM) => FALSUM)}
    |-_IPC ((p => FALSUM) => FALSUM) '&' ((q => FALSUM) => FALSUM) by Th67;
    {(p '&' q) => FALSUM,
     ((p => FALSUM) => FALSUM) '&' ((q => FALSUM) => FALSUM)}
     |-_IPC (((p => FALSUM) => FALSUM) '&' ((q => FALSUM) => FALSUM))
     => ((q => FALSUM) => FALSUM) by Th21; then
   {(p '&' q) => FALSUM,
    ((p => FALSUM) => FALSUM) '&' ((q => FALSUM) => FALSUM)}
    |-_IPC (q => FALSUM) => FALSUM by C1,Th27; then
  {(p '&' q) => FALSUM,
    ((p => FALSUM) => FALSUM) '&' ((q => FALSUM) => FALSUM)}
     |-_IPC FALSUM by B4,Th27; then
  {((p => FALSUM) => FALSUM) '&' ((q => FALSUM) => FALSUM)}
  |-_IPC ((p '&' q)=> FALSUM) => FALSUM by Th55;
  hence thesis by Th54;
end;
