reserve a,b,c for boolean object;
reserve p,q,r,s,A,B,C for Element of LTLB_WFF,
        F,G,X,Y for Subset of LTLB_WFF,
        i,j,k,n for Element of NAT,
        f,f1,f2,g for FinSequence of LTLB_WFF;
reserve M for LTLModel;

theorem Th38:
  for i,n be Nat st
  n+len f<=len f2 & (for k be Nat st 1<=k & k<=len f holds f.k=f2.(k+n)) &
  1 <=i & i<=len f holds prc f,X,i implies prc f2,X,(i+n)
 proof
  let i,n be Nat;
  assume that
   A1: n+len f<=len f2 and
   A2: for k be Nat st 1<=k & k<=len f holds f.k=f2.(k+n) and
   A3: 1<=i and
   A4: i<=len f;
  i+n<=len f+n by A4,XREAL_1:6;
  then A5: i+n<=len f2 by A1,XXREAL_0:2;
  A6: f/.i=f.i by A3,A4,Lm1
   .=f2.(i+n) by A2,A3,A4
   .=f2/.(i+n) by A3,A5,Lm1,NAT_1:12;
  assume A7: prc f,X,i;
  per cases by A7;
  suppose f.i in LTL_axioms;
   then f2.(i+n) in LTL_axioms by A2,A3,A4;
   hence prc f2,X,i+n;
  end;
  suppose f.i in X;
   then f2.(i+n) in X by A2,A3,A4;
   hence prc f2,X,i+n;
  end;
  suppose ex j,k be Nat st 1<=j & j<i & 1<=k & k<i & (f/.j,f/.k MP_rule f/.i or
f/.j,f/.k IND_rule f/.i);
   then consider j,k be Nat such that
    A8: 1<=j and
    A9: j<i and
    A10: 1<=k and
    A11: k<i and
    A12: f/.j,f/.k MP_rule f/.i or f/.j,f/.k IND_rule f/.i;
   A13: 1<=j+n & j+n<i+n by A8,A9,NAT_1:12,XREAL_1:8;
   A14: k<=len f by A4,A11,XXREAL_0:2;
   then k+n<=len f+n by XREAL_1:6;
   then A15: k+n<=len f2 by A1,XXREAL_0:2;
   A16: j<=len f by A4,A9,XXREAL_0:2;
   then j+n<=len f+n by XREAL_1:6;
   then A17: j+n<=len f2 by A1,XXREAL_0:2;
   A18: f/.k=f.k by A10,A14,Lm1
    .=f2.(k+n) by A2,A10,A14
    .=f2/.(k+n) by A10,A15,Lm1,NAT_1:12;
   A19: 1<=k+n & k+n<i+n by A10,A11,NAT_1:12,XREAL_1:8;
   f/.j=f.j by A8,A16,Lm1
    .=f2.(j+n) by A2,A8,A16
    .=f2/.(j+n) by A8,A17,Lm1,NAT_1:12;
   hence prc f2,X,i+n by A6,A12,A13,A19,A18;
  end;
  suppose ex j be Nat st 1<=j & j<i & f/.j NEX_rule f/.i;
   then consider j be Nat such that
    A20: 1<=j and
    A21: j<i and
    A22: f/.j NEX_rule f/.i;
   A23: 1<=j+n & j+n<i+n by A20,A21,NAT_1:12,XREAL_1:8;
   A24: j<=len f by A4,A21,XXREAL_0:2;
   then j+n<=len f+n by XREAL_1:6;
   then A25: j+n<=len f2 by A1,XXREAL_0:2;
   f/.j=f.j by A20,A24,Lm1
    .=f2.(j+n) by A2,A20,A24
    .=f2/.(j+n) by A20,A25,Lm1,NAT_1:12;
   hence prc f2,X,i+n by A6,A22,A23;
  end;
 end;
