reserve x,y for Real;
reserve z,z1,z2 for Complex;
reserve n for Element of NAT;

theorem
  cosh_C/.(2*z1) + cosh_C/.(2*z2) = 2*cosh_C/.(z1+z2)*cosh_C/.(z1-z2) &
  cosh_C/.(2*z1) - cosh_C/.(2*z2) = 2*sinh_C/.(z1+z2)*sinh_C/.(z1-z2)
proof
A1: 2*sinh_C/.(z1+z2)*sinh_C/.(z1-z2) = 2*(sinh_C/.(z1+z2)*sinh_C/.(z1-z2))
    .= 2*((cosh_C/.z1)*(cosh_C/.z1) - (cosh_C/.z2)*(cosh_C/.z2)) by Th59
    .= 2*(cosh_C/.z1)*(cosh_C/.z1) -1 +1- 2*((cosh_C/.z2)*(cosh_C/.z2))
    .= cosh_C/.(2*z1)+1- 2*((cosh_C/.z2)*(cosh_C/.z2)) by Th58
    .= cosh_C/.(2*z1)-(2*(cosh_C/.z2)*(cosh_C/.z2)-1);
  2*cosh_C/.(z1+z2)*cosh_C/.(z1-z2) =2*(cosh_C/.(z1+z2)*cosh_C/.(z1-z2 ))
    .=2*((sinh_C/.z1)*(sinh_C/.z1) + (cosh_C/.z2)*(cosh_C/.z2)) by Th60
    .=2*((cosh_C/.z1)*(cosh_C/.z1) - 1 + (cosh_C/.z2)*(cosh_C/.z2)) by Lm5
    .=2*(cosh_C/.z1)*(cosh_C/.z1) - 1+ (2*(cosh_C/.z2)*(cosh_C/.z2) - 1)
    .=cosh_C/.(2*z1)+ (2*(cosh_C/.z2)*(cosh_C/.z2) - 1) by Th58
    .=cosh_C/.(2*z1)+cosh_C/.(2*z2) by Th58;
  hence thesis by A1,Th58;
end;
