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
  K+`M in K+`N implies M in N
proof
  assume that
A1: K+`M in K+`N and
A2: not M in N;
  N c= M by A2,CARD_1:4;
  then K+`N c= K+`M by Th82;
  hence thesis by A1,CARD_1:4;
end;
