reserve r,r1,r2, s,x for Real,
  i for Integer;

theorem
  sin.:].PI/2,3/2*PI.[ = ].-1,1.[
proof
  sin|].PI/2,3/2*PI.[ c= sin|[.PI/2,3/2*PI.] by RELAT_1:75,XXREAL_1:25;
  then
A1: rng (sin|].PI/2,3/2*PI.[) c= rng (sin|[.PI/2,3/2*PI.]) by RELAT_1:11;
A2: rng (sin|].PI/2,3/2*PI.[) = sin.:].PI/2,3/2*PI.[ by RELAT_1:115;
  thus sin.:].PI/2,3/2*PI.[ c= ].-1,1.[
  proof
    let x be object;
    assume
A3: x in sin.:].PI/2,3/2*PI.[;
    then consider a being object such that
A4: a in dom sin and
A5: a in ].PI/2,3/2*PI.[ and
A6: sin.a = x by FUNCT_1:def 6;
    reconsider a, x as Real by A4,A6;
    set i = [\a/(2*PI)/];
A7: T(i)/(2*PI*1) = i/1 by XCMPLX_1:91;
A8: sin.a = sin a by SIN_COS:def 17;
A9: now
      assume x = 1;
      then
A10:  a = PI/2+T(i) by A6,A8,Th24;
      then PI/2 < PI/2+T(i) by A5,XXREAL_1:4;
      then PI/2-PI/2 < PI/2+T(i)-PI/2 by XREAL_1:9;
      then 0 < i;
      then
A11:  0+1 <= i by INT_1:7;
      PI/2+T(i) < 3/2*PI by A5,A10,XXREAL_1:4;
      then PI/2+T(i)-PI/2 < 3/2*PI-PI/2 by XREAL_1:9;
      then i < (1*PI)/(2*PI) by A7,XREAL_1:74;
      then i < 1/2 by XCMPLX_1:91;
      hence contradiction by A11,XXREAL_0:2;
    end;
A12: now
      assume x = -1;
      then
A13:  a = 3/2*PI+T(i) by A6,A8,Th23;
      then 3/2*PI+T(i) < 3/2*PI by A5,XXREAL_1:4;
      then 3/2*PI+T(i)-3/2*PI < 3/2*PI-3/2*PI by XREAL_1:9;
      then i < 0;
      then
A14:  i <= -1 by INT_1:8;
      PI/2 < 3/2*PI+T(i) by A5,A13,XXREAL_1:4;
      then PI/2-3/2*PI < 3/2*PI+T(i)-3/2*PI by XREAL_1:9;
      then ((-1)*PI)/(2*PI) < i by A7,XREAL_1:74;
      then (-1)/2 < i by XCMPLX_1:91;
      hence contradiction by A14,XXREAL_0:2;
    end;
    x <= 1 by A1,A2,A3,COMPTRIG:31,XXREAL_1:1;
    then
A15: x < 1 by A9,XXREAL_0:1;
    -1 <= x by A1,A2,A3,COMPTRIG:31,XXREAL_1:1;
    then -1 < x by A12,XXREAL_0:1;
    hence thesis by A15,XXREAL_1:4;
  end;
  let a be object;
  assume
A16: a in ].-1,1.[;
  then reconsider a as Real;
  -1 < a & a < 1 by A16,XXREAL_1:4;
  then a in rng (sin|[.PI/2,3/2*PI.]) by COMPTRIG:31,XXREAL_1:1;
  then consider x being object such that
A17: x in dom (sin|[.PI/2,3/2*PI.]) and
A18: (sin|[.PI/2,3/2*PI.]).x = a by FUNCT_1:def 3;
  reconsider x as Real by A17;
A19: sin.x = a by A17,A18,FUNCT_1:47;
  dom (sin|[.PI/2,3/2*PI.]) = [.PI/2,3/2*PI.] by RELAT_1:62,SIN_COS:24;
  then PI/2 <= x & x <= 3/2*PI by A17,XXREAL_1:1;
  then PI/2 < x & x < 3/2*PI or PI/2 = x or 3/2*PI = x by XXREAL_0:1;
  then x in ].PI/2,3/2*PI.[ by A16,A19,SIN_COS:76,XXREAL_1:4;
  hence thesis by A19,FUNCT_1:def 6,SIN_COS:24;
end;
