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
  cpx2euc(r*z)=r*(cpx2euc(z))
proof
A1: (cpx2euc(z))`1=Re z & (cpx2euc(z))`2=Im z by EUCLID:52;
  r = r+0 *<i>;
  then
A2: Re r = r & Im r = 0 by COMPLEX1:12;
  then
A3: Im (r*z) = r * Im z + 0 * Re z by COMPLEX1:9
    .= r * Im z;
  Re (r*z) = r * Re z - 0 * Im z by A2,COMPLEX1:9
    .= r * Re z;
  hence thesis by A3,A1,EUCLID:57;
end;
