reserve a,b,c,d,x,y,z for object, X,Y,Z for set;
reserve R,S,T for Relation;
reserve F,G for Function;

theorem Th35:
  R is well-ordering & dom F = field R & rng F c= field R & (for a
,b st [a,b] in R & a <> b holds [F.a,F.b] in R & F.a <> F.b) implies for a st a
  in field R holds [a,F.a] in R
proof
  assume that
A1: R is well-ordering & dom F = field R & rng F c= field R and
A2: [a,b] in R & a <> b implies [F.a,F.b] in R & F.a <> F.b;
  defpred P[object] means [$1,F.$1] in R;
  consider Z such that
A3: for a being object holds a in Z iff a in field R & P[a]
from XBOOLE_0:sch 1;
  now
    let a;
    assume
A4: a in field R;
    assume
A5: R-Seg(a) c= Z;
A6: now
      let b;
      assume
A7:   b in R-Seg(a);
      then
A8:  [b,F.b] in R by A3,A5;
A9:  [b,a] in R & b <> a by A7,Th1;
      then
A10:  [F.b,F.a] in R by A2;
      hence [b,F.a] in R by A1,A8,Lm2;
      F.b <> F.a by A2,A9;
      hence b <> F.a by A1,A8,A10,Lm3;
    end;
    F.a in rng F by A1,A4,FUNCT_1:def 3;
    then [a,F.a] in R by A1,A4,A6,Th34;
    hence a in Z by A3,A4;
  end;
  then
A11: field R c= Z by A1,Th33;
  let a;
  assume a in field R;
  hence thesis by A3,A11;
end;
