theorem Th82:
  |-_IPC ((p => FALSUM)'or' (q => FALSUM)) => ((p '&' q)=> FALSUM)
proof
A1: (((p => FALSUM) =>((p '&' q) => FALSUM)) '&'
  ((q => FALSUM) =>((p '&' q) => FALSUM))) =>
  (((p => FALSUM) 'or' (q => FALSUM)) => ((p '&' q) => FALSUM))
  in IPC-Taut by Th70,Th77;
A2: (p '&' q) => p in IPC-Taut by INTPRO_1:def 14;
  ((p '&' q) => p) => ((p => FALSUM) => ((p '&' q) => FALSUM)) in IPC-Taut
    by Th70,Th81; then
A3: (p => FALSUM) => ((p '&' q) => FALSUM) in IPC-Taut
    by A2,INTPRO_1:def 14;
A4: (p '&' q) => q in IPC-Taut by INTPRO_1:def 14;
  ((p '&' q) => q) => ((q => FALSUM) => ((p '&' q) => FALSUM))
  in IPC-Taut by Th70,Th81; then
   (q => FALSUM) => ((p '&' q) => FALSUM) in IPC-Taut
    by A4,INTPRO_1:def 14; then
  ((p => FALSUM) =>((p '&' q) => FALSUM)) '&'
  ((q => FALSUM) =>((p '&' q) => FALSUM)) in IPC-Taut by A3,INTPRO_1:34;
  then ((p => FALSUM) 'or' (q => FALSUM)) => ((p '&' q)=> FALSUM)
    in IPC-Taut by A1,INTPRO_1:def 14;
  hence thesis by Th69;
end;
