reserve X,Y,x,y for set;
reserve A for non empty Poset;
reserve a,a1,a2,a3,b,c for Element of A;
reserve S,T for Subset of A;
reserve f for Choice_Function of BOOL(the carrier of A);
reserve fC,fC1,fC2 for Chain of f;
reserve R for Relation,
  A for non empty Poset,
  C for Chain of A,
  S for Subset of A,
  a,a1,a2,b,c1,c2 for Element of A;

theorem Th46:
  field((the InternalRel of A) |_2 S) = S
proof
  set P = (the InternalRel of A) |_2 S;
  thus field P c= S by WELLORD1:13;
  thus S c= field P
  proof
    let x be object;
    assume
A1: x in S;
    then
A2: [x,x] in [:S,S:] by ZFMISC_1:87;
    the InternalRel of A is_reflexive_in the carrier of A by Def2;
    then [x,x] in the InternalRel of A by A1;
    then [x,x] in P by A2,XBOOLE_0:def 4;
    then x in dom P by XTUPLE_0:def 12;
    hence thesis by XBOOLE_0:def 3;
  end;
end;
