reserve a,b,m,x,y,i1,i2,i3,i for Integer,
  k,p,q,n for Nat,
  c,c1,c2 for Element of NAT,
  z for set;
reserve fp,fp1 for FinSequence of NAT,

  b,c,d, n for Element of NAT,
  a for Nat;

theorem Th35:
  for a,b st a in dom fp & b in dom fp & a<>b & len fp >= 1 holds
  fp.b divides Product Del (fp,a)
proof
  let a,b;
  assume that
A1: a in dom fp and
A2: b in dom fp and
A3: a<>b and
A4: len fp >= 1;
  consider n being Nat such that
A5: len fp = n+1 by A4,NAT_1:6;
A6: a<=n+1 by A1,A5,FINSEQ_3:25;
A7: a>=1 by A1,FINSEQ_3:25;
A8: b>=1 by A2,FINSEQ_3:25;
A9: len Del (fp,a) + 1 = n + 1 by A1,A5,WSIERP_1:def 1;
A10: b<=n+1 by A2,A5,FINSEQ_3:25;
  per cases by A6,NAT_1:8;
  suppose
A11: a<=n;
    per cases by A3,XXREAL_0:1;
    suppose
A12:  b<a;
      then b<=n by A11,XXREAL_0:2;
      then
A13:  b in dom Del (fp,a) by A8,A9,FINSEQ_3:25;
      Del (fp,a).b = fp.b by A1,A12,WSIERP_1:def 1;
      hence thesis by A13,Th32;
    end;
    suppose
A14:  a<b;
      then b>=a+1 by NAT_1:13;
      then b-1>=a by XREAL_1:19;
      then b-'1>=a by A8,XREAL_1:233;
      then
A15:  Del (fp,a).(b-'1) = fp.((b-'1)+1) by A1,WSIERP_1:def 1;
      b>1 by A7,A14,XXREAL_0:2;
      then b-1>0 by XREAL_1:50;
      then b-'1>0 by A8,XREAL_1:233;
      then
A16:  b-'1>=1 by Lm7;
      b-1<=(n+1)-1 by A10,XREAL_1:9;
      then b-'1<=n by A8,XREAL_1:233;
      then b-'1 in dom Del (fp,a) by A9,A16,FINSEQ_3:25;
      then Del (fp,a).(b-'1) divides Product Del (fp,a) by Th32;
      hence thesis by A8,A15,XREAL_1:235;
    end;
  end;
  suppose
A17: a=n+1;
    then b<n+1 by A3,A10,XXREAL_0:1;
    then b<=n by NAT_1:13;
    then
A18: b in dom Del (fp,a) by A8,A9,FINSEQ_3:25;
    b<a by A3,A10,A17,XXREAL_0:1;
    then Del (fp,a).b = fp.b by A1,WSIERP_1:def 1;
    hence thesis by A18,Th32;
  end;
end;
