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

theorem
  cosh x = 1/sqrt (1-(tanh x)^2) & sinh x = tanh(x)/sqrt (1-(tanh x)^2)
proof
A1: (sech x)^2 = (sech x)^2+(tanh x)^2-(tanh x)^2
    .= 1-(tanh x)^2 by SIN_COS5:38;
A2: sech x > 0 by Th3;
A3: cosh x = 1/(1/cosh x) by XCMPLX_1:56
    .= 1/sech(x) by SIN_COS5:def 2
    .= 1/sqrt (1-(tanh x)^2) by A1,A2,SQUARE_1:22;
  cosh x <> 0 by Lm1;
  then sinh x = sinh(x)/cosh(x)*cosh(x) by XCMPLX_1:87
    .= tanh(x)*(1/sqrt (1-(tanh x)^2)) by A3,Th1
    .= tanh(x)/sqrt (1-(tanh x)^2) by XCMPLX_1:99;
  hence thesis by A3;
end;
