let H be LTL-formula; :: thesis: for W, W1, W2 being Subset of (Subformulae H) st W = W1 \/ W2 holds
len W <= (len W1) + (len W2)
let W, W1, W2 be Subset of (Subformulae H); :: thesis: ( W = W1 \/ W2 implies len W <= (len W1) + (len W2) )
defpred S₁[ Nat] means for W, W1, W2 being Subset of (Subformulae H) st card W1 <= $1 & W = W1 \/ W2 holds
len W <= (len W1) + (len W2);
A1: S₁[ 0 ]

proof

let W, W1, W2 be Subset of (Subformulae H); :: thesis: ( card W1 <= 0 & W = W1 \/ W2 implies len W <= (len W1) + (len W2) )
assume B1: card W1 <= 0 ; :: thesis: ( not W = W1 \/ W2 or len W <= (len W1) + (len W2) )
B2: W1 = {} H by B1;
then len W1 = 0 by Thlen10;
hence ( not W = W1 \/ W2 or len W <= (len W1) + (len W2) ) by B2; :: 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 B1: S₁[k] ; :: thesis: S₁[k + 1]
S₁[k + 1]

proof

let W, W1, W2 be Subset of (Subformulae H); :: thesis: ( card W1 <= k + 1 & W = W1 \/ W2 implies len W <= (len W1) + (len W2) )
assume B2: card W1 <= k + 1 ; :: thesis: ( not W = W1 \/ W2 or len W <= (len W1) + (len W2) )
( W = W1 \/ W2 implies len W <= (len W1) + (len W2) )

proof

assume C1: W = W1 \/ W2 ; :: thesis: len W <= (len W1) + (len W2)

now

per cases ( card W1 <= k or card W1 = k + 1 ) by B2, NAT_1:8;

suppose card W1 <= k ; :: thesis: len W <= (len W1) + (len W2)

hence len W <= (len W1) + (len W2) by B1, C1; :: thesis: verum

end;

suppose card W1 = k + 1 ; :: thesis: len W <= (len W1) + (len W2)

then W1 <> {} ;
then consider x being set such that
D1: x in W1 by XBOOLE_0:def 1;
x in Subformulae H by D1;
then reconsider x = x as LTL-formula by MODELC_2:1;
set X = {x};
set Y = W1 \ {x};
set Z = (W1 \ {x}) \/ W2;
D4: {x} c= W1 by D1, ZFMISC_1:37;
then card (W1 \ {x}) = (card W1) - (card {x}) by CARD_2:63
.= (card W1) - 1 by CARD_1:50 ;
then (card (W1 \ {x})) + 1 = card W1 ;
then card (W1 \ {x}) <= k by B2, XREAL_1:8;
then ( (W1 \ {x}) \/ W2 = (W1 \ {x}) \/ W2 implies len ((W1 \ {x}) \/ W2) <= (len (W1 \ {x})) + (len W2) ) by B1;
then len ((W1 \ {x}) \/ W2) <= ((len W1) - (len x)) + (len W2) by D1, Thlen07;
then len ((W1 \ {x}) \/ W2) <= ((len W1) + (len W2)) - (len x) ;
then D5: (len ((W1 \ {x}) \/ W2)) + (len x) <= (len W1) + (len W2) by XREAL_1:21;
((W1 \ {x}) \/ W2) \/ {x} = ((W1 \ {x}) \/ {x}) \/ W2 by XBOOLE_1:4
.= W1 \/ W2 by D4, XBOOLE_1:45 ;
then len W <= (len ((W1 \ {x}) \/ W2)) + (len x) by C1, Thlen09;
hence len W <= (len W1) + (len W2) by D5, XXREAL_0:2; :: thesis: verum

end;

end;

end;

hence len W <= (len W1) + (len W2) ; :: thesis: verum

end;

hence ( not W = W1 \/ W2 or len W <= (len W1) + (len W2) ) ; :: 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 W1;
S₁[ card W1] by A3;
hence ( W = W1 \/ W2 implies len W <= (len W1) + (len W2) ) ; :: thesis: verum