reserve a,b,c,x,y,z for object,X,Y,Z for set,
  n for Nat,
  i,j for Integer,
  r,r1,r2,r3,s for Real,
  c1,c2 for Complex,
  p for Point of TOP-REAL n;

theorem Th11:
  sin(r) = sin(s) & cos(r) = cos(s) implies ex i st r = s + 2*PI*i
  proof
    assume that
A1: sin(r) = sin(s) and
A2: cos(r) = cos(s);
    consider i such that
A3: r = s + 2*PI*i or r = PI-s + 2*PI*i by A1,Th9;
    consider j such that
A4: r = s + 2*PI*j or r = -s + 2*PI*j by A2,Th10;
    per cases by A3,A4;
    suppose r = s + 2*PI*i or r = s + 2*PI*j;
      hence thesis;
    end;
    suppose r = PI-s + 2*PI*i & r = -s + 2*PI*j;
      then PI/PI = PI*(2*(j-i))/PI;
      then PI/PI = 2*(j-i) by XCMPLX_1:89;
      then 1 = 2*(j-i) by XCMPLX_1:60;
      then j-i = 1/2;
      hence thesis by NAT_D:33;
    end;
  end;
