reserve X for BCI-algebra;
reserve n for Nat;
reserve x,y for Element of X;
reserve a,b for Element of AtomSet(X);
reserve m,n for Nat;
reserve i,j for Integer;

theorem Th20:
  a=(x``)|^n implies x|^n in BranchV(a)
proof
  defpred P[Nat] means for a st a=(x``)|^$1 holds x|^$1 in BranchV(a);
A1: now
    let n;
    assume
A2: P[n];
    now
      set b=(x``)|^n;
      let a;
      assume a=(x``)|^(n+1);
      x`` in AtomSet(X) by BCIALG_1:34;
      then reconsider b as Element of AtomSet(X) by Th13;
      0.X in AtomSet(X) & x|^n in BranchV(b) by A2;
      then (x|^n)` =((x``)|^n)` by BCIALG_1:37;
      then
A3:   x|^(n+1) =x\((x``)|^n)` by Th2;
      ((x``\(((x``)|^n)`))\((x)\(((x``)|^n)` )))\(x``\x)=0.X by BCIALG_1:def 3;
      then (x``|^(n+1))\x|^(n+1)\(x``\x)=0.X by A3,Th2;
      then (x``|^(n+1))\x|^(n+1)\0.X=0.X by BCIALG_1:1;
      then (x``|^(n+1))\x|^(n+1)=0.X by BCIALG_1:2;
      then x``|^(n+1) <= x|^(n+1);
      hence a=(x``)|^(n+1) implies x|^(n+1) in BranchV(a);
    end;
    hence P[n+1];
  end;
  x|^0 = 0.X by Def1;
  then 0.X \ x|^0 = 0.X by BCIALG_1:2;
  then 0.X <= x|^0;
  then (x``)|^0 <= x|^0 by Def1;
  then
A4: P[0];
  for n holds P[n] from NAT_1:sch 2(A4,A1);
  hence thesis;
end;
