
theorem
  for q be G_INTEG st Norm(q) is Prime & Norm(q) <> 2 holds
  Re q <> 0 & Im q <> 0 & Re q <> Im q & - Re q <> Im q
  proof
    let q be G_INTEG;
    assume A1: Norm(q) is Prime & Norm(q) <> 2;
    A2: (Re q)*(Re q) = (Re q)^2
    .= |.(Re q).| ^2 by COMPLEX1:75
    .=|.(Re q).| * |.(Re q).|;
    A3: Norm(q)=(Re q+Im q*<i>)*(Re q-(Im q)*<i>) by COMPLEX1:13
    .= (Re q)^2 + (Im q)^2;
    assume A4: not (Re q <> 0 & Im q <> 0 & Re q <> Im q & - Re q <> Im q );
    per cases by A4;
    suppose Re q = 0;
      hence contradiction by A1,A3,Lm11;
    end;
    suppose Im q = 0;
      hence contradiction by A1,A3,Lm11;
    end;
    suppose A5: Re q = Im q; then
      A6: Norm(q) = 2*(Re q)*(Re q) by A3;
      |.Re q.| <> 1 by A1,A3,A5,A2;
      hence contradiction by A1,A6,Lm12;
    end;
    suppose A7: - Re q = Im q; then
      A8: Norm(q) = 2*(Re q)*(Re q) by A3;
      |.Re q.| <> 1 by A1,A3,A7,A2;
      hence contradiction by A1,A8,Lm12;
    end;
  end;
