reserve x,y,t for Real;

theorem
  1<x implies log(number_e,x)=cosh1"((x^2+1)/(2*x))
proof
  assume
A1: 1<x;
  then x<x^2 by SQUARE_1:14;
  then 1<x^2 by A1,XXREAL_0:2;
  then
A2: 1-1<x^2-1 by XREAL_1:14;
  1*2<2*x by A1,XREAL_1:68;
  then 1<2*x by XXREAL_0:2;
  then
A3: 1^2<(2*x)^2 by SQUARE_1:16;
  cosh1"((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)-1*(2*x)^2)/(2*x)
  ^2)) by A3,XCMPLX_1:126
    .=log(number_e,((x^2+1)/(2*x))+sqrt(((x^2)-1)^2)/sqrt((2*x)^2)) by A1,A2,
SQUARE_1:30
    .=log(number_e,((x^2+1)/(2*x))+(x^2-1)/sqrt((2*x)^2)) by A2,SQUARE_1:22
    .=log(number_e,((x^2+1)/(2*x))+(x^2-1)/(2*x)) by A1,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 A1,XCMPLX_1:60
    .=log(number_e,x);
  hence thesis;
end;
