reserve x,x0, r, s, h for Real,

  n for Element of NAT,
  rr, y for set,
  Z for open Subset of REAL,

  f, f1, f2 for PartFunc of REAL,REAL;

theorem Th1:
  ].-PI/2,PI/2.[ c= dom tan
proof
  ].-PI/2,PI/2.[ /\ cos"{0}={}
  proof
    assume ].-PI/2,PI/2.[ /\ cos"{0}<>{};
    then consider rr being object such that
A1: rr in ].-PI/2,PI/2.[ /\ cos"{0} by XBOOLE_0:7;
    rr in cos"{0} by A1,XBOOLE_0:def 4;
    then
A2: cos.rr in {0} by FUNCT_1:def 7;
    rr in ].-PI/2,PI/2.[ by A1,XBOOLE_0:def 4;
    then cos.rr <>0 by COMPTRIG:11;
    hence contradiction by A2,TARSKI:def 1;
  end;
  then
A3: ].-PI/2,PI/2.[ misses cos"{0} by XBOOLE_0:def 7;
  ].-PI/2,PI/2.[ \ cos"{0} c= dom cos \ cos"{0} by SIN_COS:24,XBOOLE_1:33;
  then ].-PI/2,PI/2.[ c= dom cos \ cos"{0} by A3,XBOOLE_1:83;
  then ].-PI/2,PI/2.[ c= dom sin /\ (dom cos \ cos"{0}) by SIN_COS:24
,XBOOLE_1:19;
  hence thesis by RFUNCT_1:def 1;
end;
