reserve A,B,C,D,p,q,r for Element of LTLB_WFF,
        F,G,X for Subset of LTLB_WFF,
        M for LTLModel,
        i,j,n for Element of NAT,
        f,f1,f2,g for FinSequence of LTLB_WFF;

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 prc0 f,X,i implies prc0 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: prc0 f,X,i;
    per cases by A7;
    suppose f.i in LTL0_axioms;
      hence prc0 f2,X,i+n by A2,A3,A4;
    end;
    suppose f.i in X;
      hence prc0 f2,X,i+n by A2,A3,A4;
    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 MP0_rule f/.i
      or f/.j,f/.k IND0_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 MP0_rule f/.i
      or f/.j,f/.k IND0_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 prc0 f2,X,i+n by A6,A12,A13,A19,A18;
    end;
    suppose ex j be Nat st 1<=j & j<i &
      (f/.j NEX0_rule f/.i or f/.j REFL0_rule f/.i);
      then consider j be Nat such that
A20:  1<=j and
A21:  j<i and
A22:  f/.j NEX0_rule f/.i or f/.j REFL0_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 prc0 f2,X,i+n by A6,A22,A23;
    end;
  end;
