reserve A,B for Ordinal,
  K,M,N for Cardinal,
  x,x1,x2,y,y1,y2,z,u for object,X,Y,Z,X1,X2, Y1,Y2 for set,
  f,g for Function;
reserve m,n for Nat;
reserve x1,x2,x3,x4,x5,x6,x7,x8 for object;
reserve A,B,C for Ordinal,
  K,L,M,N for Cardinal,
  x,y,y1,y2,z,u for object,X,Y,Z,Z1,Z2 for set,
  n for Nat,
  f,f1,g,h for Function,
  Q,R for Relation;
reserve n,k for Nat;

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;
