reserve x,y,t for Real;

theorem
  x>=1 implies cosh2"(x)=2*cosh2"(sqrt((x+1)/2))
proof
  assume
A1: x>=1;
  then
A2: (x-1)/2>=0 by Th7;
A3: (x^2-1)/4>=0 by A1,Th9;
A4: sqrt((x+1)/2)+sqrt((x-1)/2)>0 by A1,Th10;
  2*cosh2"(sqrt((x+1)/2)) =2*(-(log(number_e,(sqrt((x+1)/2))+sqrt(((x+1)/2
  )-1)))) by A1,SQUARE_1:def 2
    .=-(2*(log(number_e,(sqrt((x+1)/2))+sqrt((x-1)/2))))
    .=-(log(number_e,((sqrt((x+1)/2))+sqrt((x-1)/2)) to_power 2)) by A4,Lm1,
POWER:55,TAYLOR_1:11
    .=-log(number_e,((sqrt((x+1)/2))+sqrt((x-1)/2))^2) by POWER:46
    .=-log(number_e,(sqrt((x+1)/2))^2+2*(sqrt((x+1)/2))* (sqrt((x-1)/2))+(
  sqrt((x-1)/2))^2)
    .=-log(number_e,((x+1)/2)+2*(sqrt((x+1)/2))* (sqrt((x-1)/2))+(sqrt((x-1)
  /2))^2) by A1,SQUARE_1:def 2
    .=-log(number_e,(x+1)/2+2*sqrt((x+1)/2)* sqrt((x-1)/2)+(x-1)/2) by A2,
SQUARE_1:def 2
    .=-log(number_e,(x+2*(sqrt((x+1)/2)*sqrt((x-1)/2))))
    .=-log(number_e,(x+2*(sqrt(((x+1)/2)*((x-1)/2))))) by A1,A2,SQUARE_1:29
    .=-log(number_e,(x+sqrt(2^2)*(sqrt((x^2-1)/4)))) by SQUARE_1:22
    .=-log(number_e,(x+sqrt(4*((x^2-1)/4)))) by A3,SQUARE_1:29
    .=-log(number_e,(x+sqrt(x^2-1)));
  hence thesis;
end;
