reserve z,z1,z2 for Complex;
reserve r,x1,x2 for Real;
reserve p0,p,p1,p2,p3,q for Point of TOP-REAL 2;

theorem Th42:
  for p1,p2 holds |(p1,p2)| = Re ((euc2cpx(p1)) .|. (euc2cpx(p2)))
proof
  let p1,p2;
A1: p1`1=Re euc2cpx(p1) & p1`2= Im euc2cpx(p1) by COMPLEX1:12;
A2: p2`1=Re euc2cpx(p2) & p2`2= Im euc2cpx(p2) by COMPLEX1:12;
  thus |(p1,p2)| = p1`1*p2`1+p1`2*p2`2 by Th41
    .= Re ((euc2cpx(p1)) .|. (euc2cpx(p2))) by A1,A2,Th39;
end;
