reserve f for Function;
reserve p,q for FinSequence;
reserve A,B,C for set,x,x1,x2,y,z for object;
reserve k,l,m,n for Nat;
reserve a for Nat;
reserve D for non empty set;
reserve d,d1,d2,d3 for Element of D;
reserve L,M for Element of NAT;
reserve f for Function of A,B;

theorem Th66:
  card A in card B implies ex x st x in B & for y st y in A holds f.y <> x
proof
  assume that
A1: card A in card B and
A2: for x st x in B ex y st y in A & f.y = x;
A3: dom f = A by A1,CARD_1:27,FUNCT_2:def 1;
  rng f = B
  proof
    thus rng f c= B by RELAT_1:def 19;
    let x be object;
    assume x in B;
    then ex y st y in A & f.y = x by A2;
    hence thesis by A3,FUNCT_1:def 3;
  end;
  then card B c= card A by A3,CARD_1:12;
  hence thesis by A1,CARD_1:4;
end;
