reserve x, y, z, w for Real;
reserve n for Element of NAT;

theorem Th11:
  sinh(y)*sinh(z) = 1/2*(cosh(y+z)-cosh(y-z)) & sinh(y)*cosh(z) =
1/2*(sinh(y+z)+sinh(y-z)) & cosh(y)*sinh(z) = 1/2*(sinh(y+z)-sinh(y-z)) & cosh(
  y)*cosh(z) = 1/2*(cosh(y+z)+cosh(y-z))
proof
A1: sinh(y)*cosh(z) = 1/2*(sinh(y)*cosh(z)+cosh(y)*sinh(z)+(sinh(y)*cosh(z)-
  cosh(y)*sinh(z)))
    .= 1/2*(sinh(y+z)+(sinh(y)*cosh(z)-cosh(y)*sinh(z))) by Lm10
    .= 1/2*(sinh(y+z)+sinh(y-z)) by Lm10;
A2: cosh(y)*sinh(z) = 1/2*(sinh(y)*cosh(z)+cosh(y)*sinh(z)-(sinh(y)*cosh(z)-
  cosh(y)*sinh(z)))
    .= 1/2*(sinh(y+z)-(sinh(y)*cosh(z)-cosh(y)*sinh(z))) by Lm10
    .= 1/2*(sinh(y+z)-sinh(y-z)) by Lm10;
A3: cosh(y)*cosh(z) = 1/2*(cosh(y)*cosh(z)+sinh(y)*sinh(z)+(cosh(y)*cosh(z)-
  sinh(y)*sinh(z)))
    .= 1/2*(cosh(y+z)+(cosh(y)*cosh(z)-sinh(y)*sinh(z))) by Lm10
    .= 1/2*(cosh(y+z)+cosh(y-z)) by Lm10;
  sinh(y)*sinh(z) = 1/2*(cosh(y)*cosh(z)+sinh(y)*sinh(z)-(cosh(y)*cosh(z)-
  sinh(y)*sinh(z)))
    .= 1/2 * (cosh(y+z)-(cosh(y)*cosh(z)-sinh(y)*sinh(z))) by Lm10
    .= 1/2 * (cosh(y+z)-cosh(y-z)) by Lm10;
  hence thesis by A1,A2,A3;
end;
