reserve f for Function;
reserve p,q for FinSequence;
reserve A,B,C for set,x,x1,x2,y,z for object;
reserve k,l,m,n for Nat;
reserve a for Nat;
reserve D for non empty set;
reserve d,d1,d2,d3 for Element of D;
reserve L,M for Element of NAT;

theorem Th34:
  x in rng p implies len(p -| x) = x..p - 1
proof
  assume
A1: x in rng p;
  then consider n such that
A2: n = x..p - 1 and
A3: p | Seg n = p -| x by Def5;
A4: n <= n + 1 by NAT_1:12;
  n + 1 <= len p by A1,A2,Th21;
  then n <= len p by A4,XXREAL_0:2;
  hence thesis by A2,A3,FINSEQ_1:17;
end;
