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 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;
