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 x1,x2 being Real,p st x1= |.p.|*cos (Arg p) & x2=|.p.|*sin (Arg p)
  holds p = |[ x1,x2 ]|
proof
  let x1,x2 be Real,p;
  assume x1= (|.p.|)*cos (Arg(p)) & x2=(|.p.|)*sin (Arg(p));
  then x1= (|.euc2cpx(p).|)*cos (Arg(euc2cpx(p))) & x2=(|.euc2cpx(p).|)*sin (
  Arg( euc2cpx(p))) by Th25;
  then euc2cpx(p)= x1 +x2 *<i> by COMPTRIG:62;
  then p=cpx2euc(x1 +x2 *<i>) by Th2;
  hence thesis by Th5;
end;
