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

theorem
  x >= 0 implies tanh(x/2) = sqrt((cosh(x)-1)/(cosh(x)+1))
proof
  assume
A1: x >= 0;
A2: cosh(x)+1 > 0 & cosh(x)-1 >= 0 by Lm13;
  tanh(x/2) = sinh(x/2)/cosh(x/2) by Th1
    .= sqrt ((cosh(x)-1)/2)/cosh(x/2) by A1,Th24
    .= sqrt ((cosh(x)-1)/2)/sqrt((cosh(x)+1)/2) by SIN_COS5:48
    .= sqrt (((cosh(x)-1)/2)/((cosh(x)+1)/2)) by A2,SQUARE_1:30
    .= sqrt ((cosh(x)-1)/(cosh(x)+1)) by XCMPLX_1:55;
  hence thesis;
end;
