reserve x for set;
reserve i,j for Integer;
reserve n,n1,n2,n3 for Nat;
reserve p for Prime;
reserve a,b,c,d for Element of GF(p);
reserve K for Ring;
reserve a1,a2,a3,a4,a5,a6 for Element of K;

theorem Th4:
  for K being Field, a1,a2,a3,a4 being Element of K holds
  a2 <> 0.K & a4 <> 0.K & a1*a4 = a2*a3 implies a1*a2" = a3*a4"
  proof
    let K be Field, a1,a2,a3,a4 be Element of K;
    assume A1: a2 <> 0.K & a4 <> 0.K;
    assume A2: a1*a4 = a2*a3;
    a2*a3*a4" = a1*(a4"*a4) by A2,GROUP_1:def 3
      .= a1*(1.K) by A1,VECTSP_1:def 10
      .= a1;
    then a1 = (a3*a4")*a2 by GROUP_1:def 3;
    then a1*a2" = (a3*a4")*(a2"*a2) by GROUP_1:def 3
      .= (a3*a4")*(1.K) by A1,VECTSP_1:def 10
      .= a3*a4";
    hence thesis;
  end;
