reserve f for Function;
reserve p,q for FinSequence;
reserve A,B,C for set,x,x1,x2,y,z for object;
reserve k,l,m,n for Nat;
reserve a for Nat;
reserve D for non empty set;
reserve d,d1,d2,d3 for Element of D;
reserve L,M for Element of NAT;
reserve f for Function of A,B;
reserve f for Function;
reserve x1,x2,x3,x4,x5 for object;
reserve p for FinSequence;
reserve ND for non empty set;
reserve y1,y2,y3,y4,y5 for Element of ND;

theorem Th80:
  for D being non empty set, p,q being FinSequence of D st p c= q
  holds ex p9 being FinSequence of D st p ^ p9 = q
proof
  let D be non empty set, p,q be FinSequence of D;
  assume
A1: p c= q;
  dom p = Seg len p & dom q = Seg len q by FINSEQ_1:def 3;
  then Seg len p c= Seg len q by A1,GRFUNC_1:2;
  then len p <= len q by FINSEQ_1:5;
  then reconsider N = len q - len p as Element of NAT by INT_1:3,XREAL_1:48;
  defpred P[Nat,set] means q/.(len p + $1) = $2;
A2: for n being Nat st n in Seg N ex d being Element of D st P[n,d];
  consider f being FinSequence of D such that
A3: len f = N and
A4: for n being Nat st n in Seg N holds P[n,f/.n] from Sch1(A2);
  take f;
A5: len (p ^ f) = len p + N by A3,FINSEQ_1:22
    .= len q;
  now
    let k be Nat;
    assume that
A6: 1 <= k and
A7: k <= len (p ^ f);
    k in Seg len q by A5,A6,A7;
    then
A8: k in dom q by FINSEQ_1:def 3;
    per cases;
    suppose
      k <= len p;
      then k in Seg len p by A6;
      then
A9:  k in dom p by FINSEQ_1:def 3;
      hence (p ^ f).k = p.k by FINSEQ_1:def 7
        .= q.k by A1,A9,GRFUNC_1:2;
    end;
    suppose
A10:  len p < k;
      then k - len p > 0 by XREAL_1:50;
      then reconsider kk = k - len p as Element of NAT by INT_1:3;
      k <= len p + len f by A7,FINSEQ_1:22;
      then
A11:  kk <= len p + len f - len p by XREAL_1:9;
      k - len p >= 0 + 1 by A10,INT_1:7,XREAL_1:50;
      then
A12:  kk in Seg len f by A11;
      then
A13:  kk in dom f by FINSEQ_1:def 3;
      thus (p ^ f).k = f.kk by A7,A10,FINSEQ_1:24
        .= f/.kk by A13,PARTFUN1:def 6
        .= q/.(len p + kk) by A3,A4,A12
        .= q.k by A8,PARTFUN1:def 6;
    end;
  end;
  hence thesis by A5;
end;
