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
  for p st Arg p=0 holds p=|[ |.p.|,0 ]| & p`2=0
proof
  let p;
  assume Arg p=0;
  then
A1: euc2cpx(p)=|.euc2cpx(p).|+0 *<i> & Im euc2cpx(p) =0 by COMPLEX2:15,21;
  cpx2euc(|.euc2cpx(p).|+0 *<i>)=|[|.euc2cpx(p).|,0 ]| & |.euc2cpx(p).|=|.
  p.| by Th5,Th25;
  hence thesis by A1,Th2,COMPLEX1:12;
end;
