reserve X for non empty set;
reserve e for set;
reserve x for Element of X;
reserve f,g for PartFunc of X,ExtREAL;
reserve S for SigmaField of X;
reserve F for Function of RAT,S;
reserve p,q for Rational;
reserve r for Real;
reserve n,m for Nat;
reserve A,B for Element of S;

theorem
  for F being Relation st F is Finite_Sep_Sequence of S holds
  F|(Seg n) is Finite_Sep_Sequence of S
proof
  let F be Relation;
  assume
A1: F is Finite_Sep_Sequence of S;
  then reconsider G = F|(Seg n) as FinSequence of S by FINSEQ_1:18;
  reconsider F as FinSequence of S by A1;
 for k,m being object st k <> m holds G.k misses G.m
  proof
    let k,m be object;
    assume
A2: k <> m;
    per cases;
    suppose k in dom G & m in dom G;
      then G.k = F.k & G.m = F.m by FUNCT_1:47;
      hence thesis by A1,A2,PROB_2:def 2;
    end;
    suppose not (k in dom G & m in dom G);
      then G.k = {} or G.m = {} by FUNCT_1:def 2;
      hence thesis;
    end;
  end;
  hence thesis by PROB_2:def 2;
end;
