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

theorem Th45:
  cos_C/.(x+(-y)*<i>) = cos.x*cosh.y+sin.x*sinh.y*<i>
proof
  cos_C/.(x+(-y)*<i>) = cos.x*cosh.-y+(-sin.x*sinh.-y)*<i> by Th44
    .= cos.x*cosh.y+-sin.x*sinh.(-y)*<i> by SIN_COS2:19
    .= cos.x*cosh.y+-sin.x*(-sinh.y)*<i> by SIN_COS2:19
    .= cos.x*cosh.y+--sin.x*sinh.y*<i>;
  hence thesis;
end;
