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

theorem Th19:
  for x,y holds exp(x+y*<i>) = (exp_R.x)*(
  cos.y)+(exp_R.x)*(sin.y)*<i>
proof
  let x,y;
  exp(x+y*<i>)=exp_R(x)*(cos(y)+sin(y)*<i>) by Lm4
    .= exp_R(x)*cos(y)-0*sin(y) + (exp_R(x)*sin(y)+cos(y)*0)*<i>
    .=exp_R(x)*cos.y+exp_R(x)*sin(y)*<i> by SIN_COS:def 19
    .=exp_R.x*cos.y+exp_R(x)*sin(y)*<i> by SIN_COS:def 23
    .=exp_R.x*cos.y+exp_R.x*sin(y)*<i> by SIN_COS:def 23;
  hence thesis by SIN_COS:def 17;
end;
