reserve a,b,c,k,m,n for Nat;
reserve i,j,x,y for Integer;
reserve p,q for Prime;
reserve r,s for Real;

theorem
  for m being positive Nat ex a,b,c being positive Nat st
  card { [x,y] where x,y is positive Nat: a*x + b*y = c } = m
  proof
    let m be positive Nat;
    take 1, 1, m+1;
    set A = { [x,y] where x,y is positive Nat: 1*x+1*y = m+1 };
    set B = { [x,y] where x,y is positive Nat: x+y = m+1 };
    A = B
    proof
      thus A c= B
      proof
        let a be object;
        assume a in A;
        then ex x,y being positive Nat st a = [x,y] & 1*x+1*y = m+1;
        hence thesis;
      end;
      let a be object;
      assume a in B;
      then consider x,y being positive Nat such that
A1:   a = [x,y] & x+y = m+1;
      1*x+1*y = x+y;
      hence thesis by A1;
    end;
    hence thesis by Th44;
  end;
