reserve N for PT_net_Str, PTN for Petri_net, i for Nat;

theorem Th10:
  for x,y be Nat, f be FinSequence st
  f/^1 is one-to-one & 1 < x & x <= len f & 1 < y & y <= len f & f.x = f.y
  holds x = y
  proof
    let x,y be Nat, f be FinSequence;
    assume
B1: f/^1 is one-to-one & 1 < x & x <= len f & 1 < y & y <= len f & f.x = f.y;
    then
A68: 1 < len f by XXREAL_0:2;
     reconsider xm1 = x - 1,ym1 = y - 1 as Element of NAT by NAT_1:21,B1;
B8:  len (f/^1) = (len f) - 1 by RFINSEQ:def 1,A68;
B9:  x + -1 <= len f + -1 by B1,XREAL_1:6;
     1 < xm1 +1 by B1; then
     1 <= xm1 & xm1 <= len (f/^1) by NAT_1:13,B9,B8; then
B4:  xm1 in dom (f/^1) by FINSEQ_3:25;
B9a: y + -1 <= len f + -1 by B1,XREAL_1:6;
     1 < ym1 +1 by B1;then
     1 <= ym1 & ym1 <= len (f/^1) by NAT_1:13,B9a,B8; then
B5:  ym1 in dom (f/^1) by FINSEQ_3:25;
     (f/^1).xm1 = f.(xm1+1) by RFINSEQ:def 1,B4,A68
     .= f.(ym1+1) by B1 .= (f/^1).ym1 by RFINSEQ:def 1,B5,A68;then
     xm1 = ym1 by B1,FUNCT_1:def 4,B4,B5;
     hence x = y;
   end;
