theorem Th82:
  K in L & M in N or K c= L & M in N or K in L & M c= N or K c= L & M c= N
  implies K+`M c= L+`N & M+`K c= L+`N
proof
  assume that
A1: K in L & M in N or K c= L & M in N or K in L & M c= N or K c= L & M c=
  N;
A2: K c= L by A1,CARD_1:3;
A3: M c= N by A1,CARD_1:3;
A4: K+`M = card ( K +^ M);
A5: K +^ M c= L +^ N by A2,A3,ORDINAL3:18;
  hence K+`M c= L+`N by CARD_1:11;
  thus thesis by A4,A5,CARD_1:11;
end;
