reserve A,X for non empty set;
reserve f for PartFunc of [:X,X:],REAL;
reserve a for Real;

theorem Th7:
  for X being set, R being Relation of X st R is_symmetric_in X
  holds R[*] is_symmetric_in X
proof
  let X be set, R be Relation of X such that
A1: R is_symmetric_in X;
  now
    let x,y be object;
    assume that
    x in X and
    y in X and
A2: [x,y] in R[*];
A3: x in field R & y in field R by A2,FINSEQ_1:def 17;
    consider p being FinSequence such that
A4: len p >= 1 and
A5: p.1 = x and
A6: p.len p = y and
A7: for i being Nat st i >= 1 & i < len p holds [p.i,p.(i+1)] in R by A2,
FINSEQ_1:def 17;
    consider r being FinSequence such that
A8: r = Rev p;
A9: now
      let j be Nat such that
A10:  j >= 1 and
A11:  j < len r;
A12:  len p - 0 > len p - j by A10,XREAL_1:10;
      j <= len p by A8,A11,FINSEQ_5:def 3;
      then j in Seg len p by A10,FINSEQ_1:1;
      then j in dom p by FINSEQ_1:def 3;
      then
A13:  r.j = p.(len p - j + 1) by A8,FINSEQ_5:58;
A14:  j < len p by A8,A11,FINSEQ_5:def 3;
      then
A15:  len p >= j + 1 by NAT_1:13;
      j + 1 >= 1 by NAT_1:11;
      then j + 1 in Seg len p by A15,FINSEQ_1:1;
      then
A16:  j + 1 in dom p by FINSEQ_1:def 3;
      len p - j is Nat by A14,NAT_1:21;
      then len p - j in NAT by ORDINAL1:def 12;
      then consider j1 being Element of NAT such that
A17:  j1 = len p - j;
      j1 >= 1 by A15,A17,XREAL_1:19;
      then
A18:  [p.(len p - j),p.((len p - j) + 1)] in R by A7,A17,A12;
      then p.(len p - j) in X & p.(len p - j + 1) in X by ZFMISC_1:87;
      then [p.(len p - j + 1),p.(len p - (j+1) + 1)] in R by A1,A18,
RELAT_2:def 3;
      hence [r.j,r.(j+1)] in R by A8,A16,A13,FINSEQ_5:58;
    end;
A19: r.len r = r. len p by A8,FINSEQ_5:def 3
      .= x by A5,A8,FINSEQ_5:62;
    len r >= 1 & r.1 = y by A4,A6,A8,FINSEQ_5:62,def 3;
    hence [y,x] in R[*] by A3,A19,A9,FINSEQ_1:def 17;
  end;
  hence thesis by RELAT_2:def 3;
end;
