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

theorem Th26:
  sinh(2*x) = 2*sinh(x)*cosh(x) & cosh(2*x) = 2*(cosh x)^2-1 &
  tanh(2*x) = (2*tanh x)/(1+(tanh x)^2)
proof
A1: cosh(2*x) = cosh.(2*x) by SIN_COS2:def 4
    .= 2*(cosh.x)^2-1 by SIN_COS2:23
    .= 2*(cosh x)^2-1 by SIN_COS2:def 4;
A2: tanh(2*x) = tanh.(2*x) by SIN_COS2:def 6
    .= (2*tanh.x)/(1+(tanh.x)^2) by SIN_COS2:23
    .= (2*tanh x)/(1+(tanh.x)^2) by SIN_COS2:def 6
    .= (2*tanh x)/(1+(tanh x)^2) by SIN_COS2:def 6;
  sinh(2*x) = sinh.(2*x) by SIN_COS2:def 2
    .= 2*sinh.x*cosh.x by SIN_COS2:23
    .= 2*sinh(x)*cosh.x by SIN_COS2:def 2
    .= 2*sinh(x)*cosh(x) by SIN_COS2:def 4;
  hence thesis by A1,A2;
end;
