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

theorem
  x <> y & z <> y & (angle(x,y,z) = PI/2 or angle(x,y,z) = 3/2*PI)
  implies |.x-y.|^2+|.z-y.|^2 = |.x-z.|^2
proof
  assume
A1: x<>y & z<>y &( angle(x,y,z)=PI/2 or angle(x,y,z)=3/2*PI);
  set x3=x-z;
A2: |.x-y.|^2+|.z-y.|^2 =Re ((x-y).|.(x-y)) + |.z-y.|^2 by Th29
    .=Re ((x-y).|.(x-y)) + Re ((z-y).|.(z-y)) by Th29
    .=Re ((x-y).|.(x-y)+(z-y).|.(z-y)) by COMPLEX1:8;
  x3=(x-y)-(z-y);
  then
  (x-z).|.(x-z) =(x-y).|.(x-y)-(x-y).|.(z-y)-(z-y).|.(x-y) +(z-y).|.(z-y)
  by Th47;
  then Re((x-z).|.(x-z)) =Re((x-y).|.(x-y)+(z-y).|.(z-y)-((x-y).|.(z-y)+(z-y)
  .|.(x-y)))
    .=Re((x-y).|.(x-y)+(z-y).|.(z-y))-Re((x-y).|.(z-y)+(z-y).|.(x-y))by
COMPLEX1:19
    .=Re((x-y).|.(x-y)+(z-y).|.(z-y))-(Re((x-y).|.(z-y))+Re((z-y).|.(x-y)))
  by COMPLEX1:8
    .=Re((x-y).|.(x-y)+(z-y).|.(z-y))-(Re((x-y).|.(z-y))+Re(((x-y).|.(z-y))
  *')) by Th32
    .=Re((x-y).|.(x-y)+(z-y).|.(z-y))-(Re((x-y).|.(z-y))+Re((x-y).|.(z-y)))
  by COMPLEX1:27
    .=Re((x-y).|.(x-y)+(z-y).|.(z-y))-(0+Re((x-y).|.(z-y))) by A1,Lm3
    .=Re((x-y).|.(x-y)+(z-y).|.(z-y)) -0 by A1,Lm3
    .=Re((x-y).|.(x-y)+(z-y).|.(z- y));
  hence thesis by A2,Th29;
end;
