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 Th16:
  x.|.(a*y + b*z) = (a*') * x.|.y + (b*') * x.|.z
proof
  x.|.(a*y + b*z) = ((a*y + b*z).|.x)*' by Def11
    .= ( a * y.|.x + b * z.|.x )*' by Th15
    .= ( a * y.|.x )*' + ( b * z.|.x )*' by COMPLEX1:32
    .= (a*') * (y.|.x)*' + ( b * z.|.x )*' by COMPLEX1:35
    .= (a*') * (y.|.x)*' + (b*') * (z.|.x)*' by COMPLEX1:35
    .= (a*') * x.|.y + (b*') * (z.|.x)*' by Def11;
  hence thesis by Def11;
end;
