reserve X for non empty CUNITSTR;
reserve a, b for Complex;
reserve x, y for Point of X;
reserve X for ComplexUnitarySpace;
reserve x, y, z, u, v for Point of X;

theorem
  x,y are_orthogonal implies (x+y).|.(x+y) = x.|.x + y.|.y
proof
  assume
A1: x, y are_orthogonal;
  then y .|. x = 0c by Def12;
  then (x + y) .|. (x + y) = x.|.x + 0c + y.|.y by A1,Th26;
  hence thesis;
end;
