reserve a, b, c, d, x, y, z for Complex;
reserve r for Real;

theorem Th55:
  Rotate(a+b,r) = Rotate(a,r)+Rotate(b,r)
proof
  set ab = a+b;
  set rab = Rotate(ab,r),ra = Rotate(a,r), rb = Rotate(b,r);
A1: Re ab = Re a + Re b & Im ab = Im a + Im b by COMPLEX1:8;
A2: Im rab = (Re ab)*(sin r)+(Im ab)*(cos r) by Th54;
  Im ra = (Re a)*(sin r)+(Im a)*(cos r) & Im rb = (Re b)*(sin r)+(Im b)*(
  cos r ) by Th54;
  then
A3: Im (ra+rb) = (Re a)*(sin r)+(Im a)*(cos r)+((Re b)*(sin r)+(Im b)*(cos
  r)) by COMPLEX1:8
    .= Im rab by A2,A1;
A4: Re rab = (Re ab)*(cos r)-(Im ab)*(sin r) by Th54;
  Re ra = (Re a)*(cos r)-(Im a)*(sin r) & Re rb = (Re b)*(cos r)-(Im b)*(
  sin r ) by Th54;
  then
  Re (ra+rb) = (Re a)*(cos r)-(Im a)*(sin r)+((Re b)*(cos r)-(Im b)*(sin r
  )) by COMPLEX1:8
    .= Re rab by A4,A1;
  hence thesis by A3,COMPLEX1:3;
end;
