reserve x,x1,x2,y,z,z1 for set;
reserve s1,r,r1,r2 for Real;
reserve s,w1,w2 for Real;
reserve n,i for Element of NAT;
reserve X for non empty TopSpace;
reserve p,p1,p2,p3 for Point of TOP-REAL n;
reserve P for Subset of TOP-REAL n;

theorem Th7:
  i<=n implies (0*n) | i=0*i
proof
  assume
A1: i<=n;
  then i<=len (0*n) by CARD_1:def 7;
  then
A2: len ((0*n) | i)=i by FINSEQ_1:59;
A3: for j be Nat st 1<=j & j<=i holds ((0*n) | i).j=(0*i).j
  proof
    let j be Nat;
    assume that
A4: 1<=j and
A5: j<=i;
    j<=n by A1,A5,XXREAL_0:2;
    then
A6: j in Seg n by A4,FINSEQ_1:1;
A7: ((0*n) | i).j=(n|->0).j by A5,FINSEQ_3:112
      .=0 by A6,FUNCOP_1:7;
    j in Seg i by A4,A5,FINSEQ_1:1;
    hence thesis by A7,FUNCOP_1:7;
  end;
  i=len (0*i) by CARD_1:def 7;
  hence thesis by A2,A3,FINSEQ_1:14;
end;
