reserve x,y for Real;
reserve z,z1,z2 for Complex;
reserve n for Element of NAT;

theorem Th12:
  sinh_C/.(z1-z2) = (sinh_C/.z1)*(cosh_C/.z2) - (cosh_C/.z1)*( sinh_C/.z2)
proof
  sinh_C/.(z1-z2) = sinh_C/.(z1+ -z2)
    .= (sinh_C/.z1)*cosh_C/.(-z2) + (cosh_C/.z1)*sinh_C/.(-z2) by Th11
    .= (sinh_C/.z1)*cosh_C/.z2 + (cosh_C/.z1)*sinh_C/.(-z2) by Th10
    .= (sinh_C/.z1)*cosh_C/.z2 + (cosh_C/.z1)*(-sinh_C/.z2) by Th9
    .= (sinh_C/.z1)*cosh_C/.z2 + -(cosh_C/.z1)*(sinh_C/.z2);
  hence thesis;
end;
