reserve a, b, c, d, x, y, z for Complex;
reserve r for Real;

theorem Th73:
  a <> 0 & b <> 0 implies (Re (a.|.b) = 0 iff angle(a,0,b) = PI/2
  or angle(a,0,b) = 3/2*PI)
proof
A1: -Im (a.|.b)/(|.a.|*|.b.|) = (-Im (a.|.b))/(|.a.|*|.b.|);
  assume
A2: a <> 0 & b <> 0;
  then
A3: |.a.| <> 0 & |.b.| <> 0 by COMPLEX1:45;
A4: angle(a,b) = angle(a,0c,b) & 0 <= angle(a,0c,b) by Th68,Th71;
A5: angle(a,0c,b) < 2*PI & PI/2 < 2*PI by Th68,COMPTRIG:5,XXREAL_0:2;
A6: cos angle(a,b) = Re (a.|.b)/(|.a.|*|.b.|) by A2,Th67;
A7: sin angle(a,b) = - Im (a.|.b)/(|.a.|*|.b.|) by A2,Th67;
  hereby
    assume
A8: Re (a.|.b) = 0;
    then Im (a.|.b)=|.a.|*|.b.| or Im (a.|.b)= - (|.a.|*|.b.|) by Th48;
    then sin angle(a,b) = 1 or sin angle(a,b) = -1 by A7,A3,A1,XCMPLX_1:60;
    hence angle(a,0,b) = PI/2 or angle(a,0,b) = 3/2*PI by A6,A4,A5,A8,Th11,
COMPTRIG:5,SIN_COS:77;
  end;
  assume angle(a,0,b) = PI/2 or angle(a,0,b) = 3/2*PI;
  hence thesis by A6,A3,Th71,SIN_COS:77;
end;
