reserve x,y,t for Real;

theorem
  x^2<1 implies tanh"(x)=sinh"(x/sqrt(1-x^2))
proof
  assume
A1: x^2<1;
  then
A2: x+1>0 by Th11;
A3: sqrt(x+1)>0 by A1,Th11,SQUARE_1:25;
A4: (x+1)/(1-x)>0 by A1,Lm4;
A5: 1-x^2>0 by A1,XREAL_1:50;
A6: 1-x>0 by A1,Th11;
  then
A7: sqrt((x+1)/(1-x))=((x+1)/(1-x)) to_power (1/2) by A2,ASYMPT_1:83;
  sinh"(x/sqrt(1-x^2)) =log(number_e,((x/sqrt(1-x^2))+sqrt((x^2/(sqrt(1-x
  ^2))^2)+1))) by XCMPLX_1:76
    .=log(number_e,((x/sqrt(1-x^2))+sqrt((x^2/(1-x^2))+1))) by A5,
SQUARE_1:def 2
    .=log(number_e,((x/sqrt(1-x^2))+sqrt((x^2+(1-x^2)*1)/(1-x^2)))) by A5,
XCMPLX_1:113
    .=log(number_e,((x/sqrt(1-x^2))+sqrt(1)/sqrt(1-x^2))) by A5,SQUARE_1:30
    .=log(number_e,((x+1)/sqrt((1-x)*(1+x))))
    .=log(number_e,(sqrt((x+1)^2)/sqrt((1-x)*(1+x)))) by A2,SQUARE_1:22
    .=log(number_e,((sqrt(x+1)*sqrt(x+1))/sqrt((1-x)*(1+x)))) by A2,SQUARE_1:29
    .=log(number_e,((sqrt(x+1)*sqrt(x+1))/(sqrt(1-x)*sqrt(1+x)))) by A2,A6,
SQUARE_1:29
    .=log(number_e,(sqrt(x+1)/sqrt(1-x))) by A3,XCMPLX_1:91
    .=log(number_e,(sqrt((x+1)/(1-x)))) by A2,A6,SQUARE_1:30
    .=(1/2)*log(number_e,((1+x)/(1-x))) by A4,A7,Lm1,POWER:55,TAYLOR_1:11;
  hence thesis;
end;
