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

theorem
  y-z <> 0 implies (sinh(y)+sinh(z))/(cosh(y)+cosh(z)) = (cosh(y)-cosh(z
  ))/(sinh(y)-sinh(z))
proof
  assume
A1: y-z <> 0;
A2: cosh(y/2-z/2) <> 0 by Lm1;
  (sinh(y)+sinh(z))/(cosh(y)+cosh(z)) = 2*sinh(y/2+z/2)*cosh(y/2-z/2)/(
  cosh(y)+cosh(z)) by Lm11
    .= 2*sinh(y/2+z/2)*cosh(y/2-z/2)/(2*cosh(y/2+z/2)*cosh(y/2-z/2)) by Lm11
    .= 2*sinh(y/2+z/2)/(2*cosh(y/2+z/2)) * (cosh(y/2-z/2)/cosh(y/2-z/2)) by
XCMPLX_1:76
    .= 2*sinh(y/2+z/2)/(2*cosh(y/2+z/2))*1 by A2,XCMPLX_1:60
    .= 2*sinh((y+z)/2)*sinh((y-z)/2) / (2*cosh(y/2+z/2)*sinh(y/2-z/2)) by A1
,Lm5,XCMPLX_1:91
    .= 2*sinh((y-z)/2)*sinh((y+z)/2)/(2*cosh(y/2+z/2)*sinh(y/2-z/2))
    .= (cosh(y)-cosh(z))/(2*sinh(y/2-z/2)*cosh(y/2+z/2)) by Lm11
    .= (cosh(y)-cosh(z))/(sinh(y)-sinh(z)) by Lm11;
  hence thesis;
end;
