reserve p,q,r for FinSequence;
reserve u,v,x,y,y1,y2,z for object, A,D,X,Y for set;
reserve i,j,k,l,m,n for Nat;

theorem
  n in dom p iff n - 1 is Element of NAT & len p - n is Element of NAT
proof
  thus n in dom p implies n - 1 is Element of NAT & len p - n is Element of NAT
  proof
    assume
A1: n in dom p; then
A2: n <= len p by Th25;
    1 <= n by A1,Th25;
    hence thesis by A2,INT_1:5;
  end;
  assume that
A3: n - 1 is Element of NAT and
A4: len p - n is Element of NAT;
A5: 0 + n <= len p by A4,XREAL_1:19;
  0 + 1 <= n by A3,XREAL_1:19;
  hence thesis by A5,Th25;
