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 Th12:
  x.|.(y+z) = x.|.y + x.|.z
proof
  thus x.|.(y+z) = ((y+z).|.x)*' by Def11
    .= (y.|.x + z.|.x)*' by Def11
    .= (y.|.x)*' + (z.|.x)*' by COMPLEX1:32
    .= x.|.y + (z.|.x)*' by Def11
    .= x.|.y + x.|.z by Def11;
end;
