reserve x, r for Real;
reserve A for symmetrical Subset of COMPLEX;
reserve F,G for PartFunc of REAL, REAL;
reserve B for symmetrical Subset of REAL;

theorem
  (for x being Real st x in B holds sin.x<>0) implies cosec is_odd_on B
proof
  assume
A1: for x being Real st x in B holds sin.x<>0;
  B c= dom cosec
  proof
    let x be Real;
    assume
A2: x in B;
    then sin.x<>0 by A1;
    then not sin.x in {0} by TARSKI:def 1;
    then not x in sin"{0} by FUNCT_1:def 7;
    then x in dom sin \ sin"{0} by A2,SIN_COS:24,XBOOLE_0:def 5;
    hence thesis by RFUNCT_1:def 2;
  end;
  hence thesis by Th83;
end;
