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

theorem
  for x,y being Element of REAL holds (1+(-1)*<i>)/2*cosh_C/.(x+y*<i>) +
  (1+<i>)/2*cosh_C/.(x+(-y)*<i>) = sinh.x*sin.y + cosh.x*cos.y
proof
  let x,y be Element of REAL;
  set shx = sinh.x, cy = cos.y, chx = cosh.x, sy = sin.y;
  (1+(-1)*<i>)/2*cosh_C/.(x+y*<i>) + (1+<i>)/2*cosh_C/.(x+(-y)*<i>) = (1+(
  -1)*<i>)/2*(chx*cy+(shx*sy)*<i>) + (1+<i>)/2*cosh_C/.(x+(-y)*<i>) by Th48
    .= (1+(-1)*<i>)*(chx*cy+(shx*sy)*<i>)/2 + (1+<i>)/2*(chx*cy+(-shx*sy)*
  <i>) by Th49
    .= 2*(chx*cy+shx*sy+0*<i>)/2;
  hence thesis;
end;
