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 p1,p2 st p1<>0.TOP-REAL 2 & p2<>0.TOP-REAL 2 holds ( -(p1`1*p2`2)+
p1`2*p2`1= |.p1.|*|.p2.| or -(p1`1*p2`2)+p1`2*p2`1= -(|.p1.|*|.p2.|) iff angle(
  p1,0.TOP-REAL 2,p2)=PI/2 or angle(p1,0.TOP-REAL 2,p2)=3/2*PI)
proof
  let p1,p2;
A1: p2`1=Re euc2cpx(p2) & p2`2= Im euc2cpx(p2) by COMPLEX1:12;
  p1`1=Re euc2cpx(p1) & p1`2= Im euc2cpx(p1) by COMPLEX1:12;
  then
A2: Im ((euc2cpx(p1)) .|. (euc2cpx(p2))) = -(p1`1*p2`2)+p1`2*p2`1 by A1,Th40;
  assume p1<>0.TOP-REAL 2 & p2<>0.TOP-REAL 2;
  then
A3: euc2cpx(p1)<> 0c & euc2cpx(p2)<> 0c by Th2,Th16;
  |.euc2cpx(p1).|=|.p1.| & |.euc2cpx(p2).|=|.p2.| by Th25;
  hence thesis by A3,A2,Th17,COMPLEX2:76;
end;
