reserve i,j for Nat;
reserve x,y for set;
reserve A for non empty set;
reserve c for Element of StandardStackSystem A;
reserve m for stack of StandardStackSystem A;
reserve X for non empty non void StackSystem;
reserve s,s1,s2 for stack of X;
reserve e,e1,e2 for Element of X;
reserve X for StackAlgebra;
reserve s,s1,s2,s3 for stack of X;
reserve e,e1,e2,e3 for Element of X;

theorem Th23:
  for R being Relation of A
  for t being RedSequence of R holds t.1 in A iff t is A-valued
  proof
    let R be Relation of A;
    let t be RedSequence of R;
    rng t <> {}; then
    1 in dom t by FINSEQ_3:32; then
A1: t.1 in rng t by FUNCT_1:def 3;
    hereby
      assume
A2:   t.1 in A;
      defpred P[Nat] means $1 in dom t implies t.$1 in A;
A3:   P[0] by FINSEQ_3:24;
A4:   P[i] implies P[i+1]
      proof assume
        P[i]; assume
A5:     i+1 in dom t & t.(i+1) nin A;
        i = 0 or i >= 0+1 by NAT_1:13; then
        consider j being Nat such that
A6:     i = j+1 by A2,A5,NAT_1:6;
        i <= i+1 & i+1 <= len t by A5,FINSEQ_3:25,NAT_1:11; then
        1 <= i & i <= len t by A6,NAT_1:11,XXREAL_0:2; then
        i in dom t by FINSEQ_3:25; then
        [t.i, t.(i+1)] in R by A5,REWRITE1:def 2;
        hence thesis by A5,ZFMISC_1:87;
      end;
A7:   P[i] from NAT_1:sch 2(A3,A4);
      thus t is A-valued
      proof
        let x be object; assume x in rng t; then
        consider y being object such that
A8:     y in dom t & x = t.y by FUNCT_1:def 3;
        reconsider y as Nat by A8;
        thus thesis by A8,A7;
      end;
    end;
    assume rng t c= A;
    hence t.1 in A by A1;
  end;
