reserve x,y for set,
  i,j,k,l,m,n for Nat,
  K for Field,
  N for without_zero finite Subset of NAT,
  a,b for Element of K,
  A,B,B1,B2,X,X1,X2 for (Matrix of K),
  A9 for (Matrix of m,n,K),
  B9 for (Matrix of m,k,K);

theorem Th8:
  i in Seg m implies Sgm (Seg (n+m)\Seg n).i=n+i
proof
  assume
A1: i in Seg m;
  reconsider N=n as Element of NAT by ORDINAL1:def 12;
  set I=idseq m;
A2: dom I=Seg len I by FINSEQ_1:def 3;
A3: dom (N Shift I) = {k+N where k is Nat: k in dom I} by VALUED_1:def 12;
A4: Seg (n+m)\Seg n c= dom (N Shift I)
  proof
    let x be object such that
A5: x in Seg (n+m)\Seg n;
    reconsider i=x as Element of NAT by A5;
A6: i in Seg (n+m) by A5,XBOOLE_0:def 5;
    not i in Seg n by A5,XBOOLE_0:def 5;
    then
A7: i<1 or i>n;
    then reconsider IN=i-n as Element of NAT by A6,FINSEQ_1:1,NAT_1:21;
A8: n+IN=i;
    i<=n+m by A6,FINSEQ_1:1;
    then
A9: IN<=m by A8,XREAL_1:8;
    IN>=1 by A6,A7,A8,FINSEQ_1:1,NAT_1:19;
    then IN in dom I by A9;
    then n+IN in dom (N Shift I) by A3;
    hence thesis;
  end;
  dom (N Shift I) c=Seg (n+m)\Seg n
  proof
    let x be object;
    assume x in dom (N Shift I);
    then consider k be Nat such that
A10: k+n=x and
A11: k in dom I by A3;
    k<=m by A11,FINSEQ_1:1;
    then
A12: n+k<=n+m by XREAL_1:7;
    1<=k by A11,FINSEQ_1:1;
    then
A13: n+1<=n+k by XREAL_1:7;
    then n+k >n by NAT_1:13;
    then
A14: not k+n in Seg n by FINSEQ_1:1;
    1<=n+1 by NAT_1:11;
    then 1<=n+k by A13,XXREAL_0:2;
    then n+k in Seg (n+m) by A12;
    hence thesis by A10,A14,XBOOLE_0:def 5;
  end;
  then Seg (n+m)\Seg n = dom (N Shift I) by A4;
  hence thesis by A1,A2,VALUED_1:44;
end;
