
theorem Th18:
  for A being non-empty partial UAStr
  for R being Equivalence_Relation of the carrier of A st R c= DomRel A
  holds R|^A is total symmetric transitive
proof
  let A be non-empty partial UAStr;
  let R be Equivalence_Relation of the carrier of A;
  assume
A1: R c= DomRel A;
  now
    let x be object;
    assume x in the carrier of A;
    then reconsider a = x as Element of A;
A2: [a,a] in R by EQREL_1:5;
    now
      let f be operation of A, p,q be FinSequence;
      assume that
A3:   p^<*a*>^q in dom f and p^<*a*>^q in dom f;
      f.(p^<*a*>^q) in rng f by A3,FUNCT_1:def 3;
      hence [f.(p^<*a*>^q), f.(p^<*a*>^q)] in R by EQREL_1:5;
    end;
    hence [x,x] in R|^A by A2,Def5;
  end;
  then
A4: R|^A is_reflexive_in the carrier of A;
  then
A5: dom(R|^A) = the carrier of A by ORDERS_1:13;
A6: field(R|^A) = the carrier of A by A4,ORDERS_1:13;
  thus R|^A is total by A5,PARTFUN1:def 2;
  now
    let x,y be object;
    assume that
A7: x in the carrier of A and
A8: y in the carrier of A;
    reconsider a = x, b = y as Element of A by A7,A8;
    assume
A9: [x,y] in R|^A;
    then
A10: [a,b] in R by Def5;
    now
      thus [b,a] in R by A10,EQREL_1:6;
      let f be operation of A;
      let p,q be FinSequence;
      assume that
A11:  p^<*b*>^q in dom f and
A12:  p^<*a*>^q in dom f;
      [f.(p^<*a*>^q), f.(p^<*b*>^q)] in R by A9,A11,A12,Def5;
      hence [f.(p^<*b*>^q), f.(p^<*a*>^q)] in R by EQREL_1:6;
    end;
    hence [y,x] in R|^A by Def5;
  end;
  then R|^A is_symmetric_in the carrier of A;
  hence R|^A is symmetric by A6;
  now
    let x,y,z be object;
    assume that
A13: x in the carrier of A and
A14: y in the carrier of A and
A15: z in the carrier of A;
    reconsider a = x, b = y, c = z as Element of A by A13,A14,A15;
    assume that
A16: [x,y] in R|^A and
A17: [y,z] in R|^A;
A18: now
      let f be operation of A;
      let p,q be FinSequence;
A19:  [a,b] in R by A16,Def5;
A20:  p^<*a*>^q in dom f & p^<*b*>^q in dom f implies [f.(p^<*a*>^q), f.(p^
      <*b*>^q)] in R by A16,Def5;
      p^<*b*>^q in dom f & p^<*c*>^q in dom f implies [f.(p^<*b*>^q), f.(p^
      <*c*>^q)] in R by A17,Def5;
      hence p^<*a*>^q in dom f & p^<*c*>^q in dom f
      implies [f.(p^<*a*>^q), f.(p^<*c*>^q)] in R by A1,A19,A20,Def4,EQREL_1:7;
    end;
A21: [a,b] in R by A16,Def5;
    [b,c] in R by A17,Def5;
    then [a,c] in R by A21,EQREL_1:7;
    hence [x,z] in R|^A by A18,Def5;
  end;
  then R|^A is_transitive_in the carrier of A;
  hence thesis by A6;
end;
