reserve x,y,t for Real;

theorem
  0<x & x<1 & 1<=(x^2+1)/(2*x) implies log(number_e,x)=cosh2"((x^2+1)/(2 *x))
proof
  assume that
A1: 0<x and
A2: x<1 and
 1<=(x^2+1)/(2*x);
A3: (1/x)=x to_power (-1) by A1,Th1;
  x^2<x by A1,A2,SQUARE_1:13;
  then x^2<1 by A2,XXREAL_0:2;
  then
A4: x^2-x^2<1-x^2 by XREAL_1:14;
  0*2<x*2 by A1;
  then
A5: 0<(2*x)^2;
  cosh2"((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 A5,XCMPLX_1:126
    .=-log(number_e,((x^2+1)/(2*x))+sqrt((1-(x^2))^2)/sqrt(2*x)^2) by A1,A4,
SQUARE_1:30
    .=-log(number_e,((x^2+1)/(2*x))+(1-(x^2))/sqrt(2*x)^2) by A4,SQUARE_1:22
    .=-log(number_e,((x^2+1)/(2*x))+(1-x^2)/(2*x)) by A1,SQUARE_1:22
    .=-log(number_e,((x^2+1)+(1-x^2))/(2*x))
    .=-log(number_e,(2*1)/(2*x))
    .=-log(number_e,1/x) by XCMPLX_1:91
    .=-((-1)*log(number_e,x)) by A1,A3,Lm1,POWER:55,TAYLOR_1:11
    .=log(number_e,x);
  hence thesis;
end;
