
theorem Th20:
  for A being non-empty partial UAStr
  for R being Equivalence_Relation of the carrier of A st R c= DomRel A
  for i being Element of NAT holds R|^(A,i) is total symmetric transitive
proof
  let A be non-empty partial UAStr;
  let R be Equivalence_Relation of the carrier of A such that
A1: R c= DomRel A;
  defpred P[Nat] means
  R|^(A,$1) c= DomRel A & R|^(A,$1) is total symmetric transitive;
A2: P[ 0 ] by A1,Th15;
A3: now
    let i be Nat;
    assume
A4: P[i];
A5: R|^(A,i)|^A c= R|^(A,i) by Th19;
    R|^(A,i)|^A = R|^(A,i+1) by Th16;
    hence P[i+1] by A4,A5,Th18,XBOOLE_1:1;
  end;
  for i being Nat holds P[i] from NAT_1:sch 2(A2,A3);
  hence thesis;
end;
