theorem Th127:
  |-_IPC (p => q) => (((q '&' r)=> FALSUM) => ((p '&' r)=> FALSUM))
proof
   p '&' r in {p '&' r,(q '&' r)=> FALSUM,p => q} by ENUMSET1:def 1; then
A1: {p '&' r,(q '&' r)=> FALSUM,p => q} |-_IPC p '&' r by Th67;
   (q '&' r)=> FALSUM in {p '&' r,(q '&' r)=> FALSUM,p => q}
       by ENUMSET1:def 1; then
A2: {p '&' r,(q '&' r)=> FALSUM,p => q} |-_IPC (q '&' r)=> FALSUM by Th67;
   p => q in {p '&' r,(q '&' r)=> FALSUM,p => q} by ENUMSET1:def 1; then
A3: {p '&' r,(q '&' r)=> FALSUM,p => q} |-_IPC p => q by Th67;
   {p '&' r,(q '&' r)=> FALSUM,p => q} |-_IPC (p '&' r) => r by Th21; then
A5: {p '&' r,(q '&' r)=> FALSUM,p => q} |-_IPC r by A1,Th27;
   {p '&' r,(q '&' r)=> FALSUM,p => q} |-_IPC (p '&' r) => p by Th20; then
   {p '&' r,(q '&' r)=> FALSUM,p => q} |-_IPC p by A1,Th27; then
A6: {p '&' r,(q '&' r)=> FALSUM,p => q} |-_IPC q by A3,Th27;
   {p '&' r,(q '&' r)=> FALSUM,p => q} |-_IPC q => (r => (q '&' r))
       by Th22; then
   {p '&' r,(q '&' r)=> FALSUM,p => q} |-_IPC r => (q '&' r)
       by A6,Th27; then
   {p '&' r,(q '&' r)=> FALSUM,p => q} |-_IPC q '&' r by A5,Th27; then
   {p '&' r,(q '&' r)=> FALSUM,p => q} |-_IPC FALSUM by A2,Th27; then
   {(q '&' r)=> FALSUM,p => q} |-_IPC (p '&' r)=> FALSUM by Th56; then
  {p => q} |-_IPC ((q '&' r)=> FALSUM) => ((p '&' r)=> FALSUM) by Th55;
  hence thesis by Th54;
end;
