
theorem
for a being Complex holds a * a*' = |.a.|^2
proof
let a be Complex;
set ac = a;
reconsider b = a*' as Complex;
reconsider Ra = Re ac, Ia = Im ac as Real;
reconsider Ra2 = Ra^2, Ia2 = Ia^2 as Real;
A1: |.a.| = sqrt(Ra2 + Ia2) by COMPLEX1:def 12;
reconsider xx = Ra2 + Ia2 as Real;
Ra2 >= 0 & 0 <= Ia2 by XREAL_1:63;
then |.a.|^2 = Re ac * Re ac - Im ac * (- Im ac) +
               (Re ac * (- Im ac) + Re ac * Im ac) * <i> by A1,SQUARE_1:def 2
            .= Re ac * Re ac - Im ac * (- Im ac) +
               (Re ac * Im b + Re ac * Im ac) * <i> by COMPLEX1:27
            .= Re ac * Re ac - Im ac * Im b +
               (Re ac * Im b + Re ac * Im ac) * <i> by COMPLEX1:27
            .= Re ac * Re ac - Im ac * Im b +
               (Re ac * Im b + Re b * Im ac) * <i> by COMPLEX1:27
            .= Re ac * Re b - Im ac * Im b +
               (Re ac * Im b + Re b * Im ac) * <i> by COMPLEX1:27
            .= ac * b by COMPLEX1:82;
hence thesis;
end;
