let L1, L2 be FinSequence; :: thesis: for H being LTL-formula
for W being Subset of (Subformulae H) st rng L1 = Subformulae H & L1 is one-to-one & rng L2 = Subformulae H & L2 is one-to-one holds
len L1,W = len L2,W
let H be LTL-formula; :: thesis: for W being Subset of (Subformulae H) st rng L1 = Subformulae H & L1 is one-to-one & rng L2 = Subformulae H & L2 is one-to-one holds
len L1,W = len L2,W
let W be Subset of (Subformulae H); :: thesis: ( rng L1 = Subformulae H & L1 is one-to-one & rng L2 = Subformulae H & L2 is one-to-one implies len L1,W = len L2,W )
defpred S₁[ Nat] means for W1 being Subset of (Subformulae H) st card W1 <= $1 & rng L1 = Subformulae H & L1 is one-to-one & rng L2 = Subformulae H & L2 is one-to-one holds
len L1,W1 = len L2,W1;
A1: S₁[ 0 ]

proof

let W1 be Subset of (Subformulae H); :: thesis: ( card W1 <= 0 & rng L1 = Subformulae H & L1 is one-to-one & rng L2 = Subformulae H & L2 is one-to-one implies len L1,W1 = len L2,W1 )
assume B1: card W1 <= 0 ; :: thesis: ( not rng L1 = Subformulae H or not L1 is one-to-one or not rng L2 = Subformulae H or not L2 is one-to-one or len L1,W1 = len L2,W1 )
B2: W1 = {} H by B1;
then len L1,W1 = 0 by Thlen00;
hence ( not rng L1 = Subformulae H or not L1 is one-to-one or not rng L2 = Subformulae H or not L2 is one-to-one or len L1,W1 = len L2,W1 ) by B2, Thlen00; :: thesis: verum

end;

A2: for k being Nat st S₁[k] holds
S₁[k + 1]

proof

let k be Nat; :: thesis: ( S₁[k] implies S₁[k + 1] )
assume B0: S₁[k] ; :: thesis: S₁[k + 1]
S₁[k + 1]

proof

let W1 be Subset of (Subformulae H); :: thesis: ( card W1 <= k + 1 & rng L1 = Subformulae H & L1 is one-to-one & rng L2 = Subformulae H & L2 is one-to-one implies len L1,W1 = len L2,W1 )
assume B1: card W1 <= k + 1 ; :: thesis: ( not rng L1 = Subformulae H or not L1 is one-to-one or not rng L2 = Subformulae H or not L2 is one-to-one or len L1,W1 = len L2,W1 )
( rng L1 = Subformulae H & L1 is one-to-one & rng L2 = Subformulae H & L2 is one-to-one implies len L1,W1 = len L2,W1 )

proof

assume C1: ( rng L1 = Subformulae H & L1 is one-to-one & rng L2 = Subformulae H & L2 is one-to-one ) ; :: thesis: len L1,W1 = len L2,W1

now

per cases ( card W1 <= k or card W1 = k + 1 ) by B1, NAT_1:8;

suppose card W1 <= k ; :: thesis: len L1,W1 = len L2,W1

hence len L1,W1 = len L2,W1 by B0, C1; :: thesis: verum

end;

suppose E1: card W1 = k + 1 ; :: thesis: len L1,W1 = len L2,W1

then W1 <> {} ;
then consider F being set such that
E2: F in W1 by XBOOLE_0:def 1;
F in Subformulae H by E2;
then reconsider F = F as LTL-formula by MODELC_2:1;
{F} c= W1 by E2, ZFMISC_1:37;
then E3: card (W1 \ {F}) = (card W1) - (card {F}) by CARD_2:63
.= (card W1) - 1 by CARD_1:50
.= k by E1 ;
E4: len L1,W1 = ((len L1,W1) - (len F)) + (len F)
.= (len L1,(W1 \ {F})) + (len F) by C1, E2, Thlen02 ;
len L2,W1 = ((len L2,W1) - (len F)) + (len F)
.= (len L2,(W1 \ {F})) + (len F) by C1, E2, Thlen02
.= (len L1,(W1 \ {F})) + (len F) by E3, B0, C1 ;
hence len L1,W1 = len L2,W1 by E4; :: thesis: verum

end;

end;

end;

hence len L1,W1 = len L2,W1 ; :: thesis: verum

end;

hence ( not rng L1 = Subformulae H or not L1 is one-to-one or not rng L2 = Subformulae H or not L2 is one-to-one or len L1,W1 = len L2,W1 ) ; :: thesis: verum

end;

hence S₁[k + 1] ; :: thesis: verum

end;

A3: for k being Nat holds S₁[k] from NAT_1:sch 2(A1, A2);
set k = card W;
S₁[ card W] by A3;
hence ( rng L1 = Subformulae H & L1 is one-to-one & rng L2 = Subformulae H & L2 is one-to-one implies len L1,W = len L2,W ) ; :: thesis: verum