reserve x,y,t for Real;

theorem
  x^2<1 & y^2<1 implies tanh"(x)-tanh"(y)=tanh"((x-y)/(1-x*y))
proof
  assume
A1: x^2<1 & y^2<1;
  then
A2: 0<(1+x)/(1-x) & 0<(1+y)/(1-y) by Lm4;
A3: tanh"(x)-tanh"(y) =(1/2)*(log(number_e,(1+x)/(1-x))-log(number_e,(1+y)/(
  1-y)))
    .=(1/2)*(log(number_e,((1+x)/(1-x))/((1+y)/(1-y)))) by A2,Lm1,POWER:54
,TAYLOR_1:11
    .=(1/2)*(log(number_e,((1+x)*(1-y))/((1-x)*(1+y)))) by XCMPLX_1:84
    .=(1/2)*(log(number_e,(1-y+x-x*y)/(1+y-x-x*y)));
A4: 1-x*y<>0 by A1,Th28;
  then
  tanh"((x-y)/(1-x*y)) =(1/2)*log(number_e,(((x-y)+(1-x*y)*1)/(1-x*y))/(1-
  ((x-y)/(1-x*y)))) by XCMPLX_1:113
    .=(1/2)*log(number_e,((x-y+1-x*y)/(1-x*y))/((1*(1-x*y)-(x-y))/(1-x*y)))
  by A4,XCMPLX_1:127
    .=(1/2)*(log(number_e,(1-y+x-x*y)/(1+y-x-x*y))) by A4,XCMPLX_1:55;
  hence thesis by A3;
end;
