reserve p,q,r,th,th1 for Real;
reserve n for Nat;

theorem Th18:
  (sinh.p)^2 = 1/2*(cosh.(2*p) - 1) & (cosh.p)^2 = 1/2*(cosh.(2*p) + 1)
proof
A1: (cosh.p)^2 =( (exp_R.(p) + exp_R.(-p))/2 )*(cosh.p) by Def3
    .=( (exp_R.(p) + exp_R.(-p))/2 )*( (exp_R.(p) + exp_R.(-p))/2 ) by Def3
    .=( (exp_R.(p))*(exp_R.(p))+(exp_R.(p))*(exp_R.(-p)) +(exp_R.(-p))*(
  exp_R.(p))+(exp_R.(-p))*(exp_R.(-p)) )/4
    .=( (exp_R.(p+p))+(exp_R.(p))*(exp_R.(-p)) +(exp_R.(p))*(exp_R.(-p))+(
  exp_R.(-p))*(exp_R.(-p)) )/4 by Th12
    .=( (exp_R.(p+p))+(exp_R.(p))*(exp_R.(-p)) +(exp_R.(p))*(exp_R.(-p))+(
  exp_R.(-p+ -p)) )/4 by Th12
    .=( (exp_R.(p+p))+(exp_R.(p+ -p)) +(exp_R.(p))*(exp_R.(-p))+(exp_R.(-p+
  -p)) )/4 by Th12
    .=( (exp_R.(p+p))+(exp_R.(p+ -p)) +(exp_R.(p+ -p))+(exp_R.(-p+ -p)) )/4
  by Th12
    .=( exp_R.(2*p) + 2*(exp_R.(0)) +exp_R.(-p+-p) )/4
    .=( exp_R.(2*p) + 2*1 +exp_R.(-(2*p)) )/4 by SIN_COS:51,def 23
    .=1/2*( ( exp_R.(2*p) +exp_R.(-(2*p)) )/2 ) + (1*2)/(2*2)
    .=1/2*(cosh.(2*p))+ 1/2*((2*1/2)) by Def3
    .=1/2*( cosh.(2*p)+ 1 );
  (sinh.p)^2 =( (exp_R.(p) - exp_R.(-p))/2 )*(sinh.p) by Def1
    .=( (exp_R.(p) - exp_R.(-p))/2 )*( (exp_R.(p) - exp_R.(-p))/2 ) by Def1
    .=( (exp_R.(p))*(exp_R.(p))-(exp_R.(p))*(exp_R.(-p)) -(exp_R.(-p))*(
  exp_R.(p))+(exp_R.(-p))*(exp_R.(-p)) )/4
    .=( (exp_R.(p+p))-(exp_R.(p))*(exp_R.(-p)) -(exp_R.(p))*(exp_R.(-p))+(
  exp_R.(-p))*(exp_R.(-p)) )/4 by Th12
    .=( (exp_R.(p+p))-(exp_R.(p))*(exp_R.(-p)) -(exp_R.(p))*(exp_R.(-p))+(
  exp_R.(-p+ -p)) )/4 by Th12
    .=( (exp_R.(p+p))-(exp_R.(p+ -p)) -(exp_R.(p))*(exp_R.(-p))+(exp_R.(-p+
  -p)) )/4 by Th12
    .=( ((exp_R.(p+p))+ -(exp_R.(p+ -p)) -(exp_R.(p+ -p)) )+(exp_R.(-p+ -p))
  )/4 by Th12
    .=( exp_R.(2*p) + 2*(-(exp_R.(0))) +exp_R.(-p+-p) )/4
    .=( exp_R.(2*p) + 2*(-1) +exp_R.(-(2*p)) )/4 by SIN_COS:51,def 23
    .=1/2*( ( exp_R.(2*p) +exp_R.(-(2*p)) )/2 ) + ((-1)*2)/(2*2)
    .=1/2*(cosh.(2*p))+ 1/2*((2*(-1))/2) by Def3
    .=1/2*( cosh.(2*p) - 1 );
  hence thesis by A1;
end;
