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

theorem
  x <> 0 implies exp_R(x) = (coth(x/2)+1)/(coth(x/2)-1) & exp_R(-x) = (
  coth(x/2)-1)/(coth(x/2)+1)
proof
  assume x <> 0;
  then
A1: x/2 <> 0;
A2: coth(x/2)-1 = 1/(1/coth(x/2))-1 by XCMPLX_1:56
    .= 1/tanh(x/2)-1 by Th2
    .= 1/tanh(x/2)-tanh(x/2)/tanh(x/2) by A1,Lm4,XCMPLX_1:60
    .= (1-tanh(x/2))/tanh(x/2) by XCMPLX_1:120;
A3: coth(x/2)+1 = 1/(1/coth(x/2))+1 by XCMPLX_1:56
    .= 1/tanh(x/2)+1 by Th2
    .= 1/tanh(x/2)+tanh(x/2)/tanh(x/2) by A1,Lm4,XCMPLX_1:60
    .= (1+tanh(x/2))/tanh(x/2) by XCMPLX_1:62;
A4: exp_R(-x) = (1-tanh(x/2))/(1+tanh(x/2)) by Th7
    .= (coth(x/2)-1)/(coth(x/2)+1)by A1,A3,A2,Lm4,XCMPLX_1:55;
  exp_R(x) = (1+tanh(x/2))/(1-tanh(x/2)) by Th7
    .= (coth(x/2)+1)/(coth(x/2)-1)by A1,A3,A2,Lm4,XCMPLX_1:55;
  hence thesis by A4;
end;
