reserve x,y,t for Real;

theorem
  0<x implies log(number_e,x)=sinh"((x^2-1)/(2*x))
proof
A1: x^2>=0 by XREAL_1:63;
  assume
A2: 0<x;
  then 0*2<x*2;
  then
A3: 0<(2*x)^2;
  sinh"((x^2-1)/(2*x)) =log(number_e,(((x^2-1)/(2*x))+sqrt((x^2-1)^2/(2*x)
  ^2+1))) by XCMPLX_1:76
    .=log(number_e,(((x^2-1)/(2*x))+sqrt((((x^2)^2-2*x^2+1)+ (2*x)^2*1)/(2*x
  )^2))) by A3,XCMPLX_1:113
    .=log(number_e,(((x^2-1)/(2*x))+sqrt((x^2+1)^2)/sqrt((2*x)^2))) by A2,
SQUARE_1:30
    .=log(number_e,(((x^2-1)/(2*x))+(x^2+1)/sqrt((2*x)^2))) by A1,SQUARE_1:22
    .=log(number_e,((x^2-1)/(2*x)+(x^2+1)/(2*x))) by A2,SQUARE_1:22
    .=log(number_e,(((x^2-1)+(x^2+1))/(2*x)))
    .=log(number_e,((2*x^2)/(2*x)))
    .=log(number_e,(x*x)/x) by XCMPLX_1:91
    .=log(number_e,x/(x/x)) by XCMPLX_1:77
    .=log(number_e,x/1) by A2,XCMPLX_1:60
    .=log(number_e,x);
  hence thesis;
end;
