 reserve i,j, k,v, w for Nat;
 reserve j1,j2, m, n, s, t, x, y for Integer;
 reserve p for odd Prime;
 reserve a for Real;
 reserve b for Integer;

theorem lem10:
  for x1,h be Nat, y1 be Integer st 1 < h & x1 mod h = y1 mod h
    & y1 = 0 holds ex m1 be Integer st x1 = h*m1
  proof
    let x1, h be Nat, y1 be Integer;
    assume that
A1: 1 < h and
A2: x1 mod h = y1 mod h and
A3: y1 = 0;
A5: x1 mod h = 0 by NAT_D:26,A2,A3;
    reconsider x1 as Integer;
A6: h divides x1 by A1,A5,INT_1:62;
    reconsider h as Integer;
    thus thesis by A6,INT_1:def 3;
  end;
