reserve A,B,a,b,c,d,e,f,g,h for set;

theorem Th27:
  for R be symmetric RelStr,x,y being set holds the InternalRel of
  R reduces x,y implies the InternalRel of R reduces y,x
proof
  let R be symmetric RelStr;
  set IR = the InternalRel of R;
  let x,y be set;
A1: IR = IR~ by RELAT_2:13;
  assume IR reduces x,y;
  then consider p being RedSequence of IR such that
A2: p.1 = x and
A3: p.len p = y by REWRITE1:def 3;
  reconsider p as FinSequence;
A4: (Rev p).len p = x by A2,FINSEQ_5:62;
  IR reduces y,x
  proof
    reconsider q = Rev p as RedSequence of IR by A1,REWRITE1:9;
    q.1 = y & q.len q = x by A3,A4,FINSEQ_5:62,def 3;
    hence thesis by REWRITE1:def 3;
  end;
  hence thesis;
end;
