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

theorem Th47:
  sinh_C/.(x+(-y)*<i>) = sinh.x*cos.y+(-cosh.x*sin.y)*<i>
proof
  sinh_C/.(x+(-y)*<i>) = sinh.x*cos.-y+cosh.x*sin.(-y)*<i> by Th46
    .= sinh.x*cos.y+cosh.x*sin.(-y)*<i> by SIN_COS:30
    .= sinh.x*cos.y+cosh.x*(-sin.y)*<i> by SIN_COS:30;
  hence thesis;
end;
