theorem Th75:
  not M is finite & (N c= M or N in M) implies M+`N = M & N+`M = M
proof
  assume that
A1: not M is finite and
A2: N c= M or N in M;
A3: M+`M = M by A1,Th74;
  N c= M by A2,CARD_1:3;
  then
A4: M +^ N c= M +^ M by ORDINAL2:33;
A5: M c= M +^ N by ORDINAL3:24;
A6: card M = M;
A7: M+`N c= M by A3,A4,CARD_1:11;
  M c= M+`N by A5,A6,CARD_1:11;
  hence thesis by A7;
end;
