let IT1, IT2 be AP:VLabelingSeq of EL; :: thesis: ( IT1 . 0 = source .--> 1 & ( for n being Nat holds IT1 . (n + 1) = AP:Step (IT1 . n) ) & IT2 . 0 = source .--> 1 & ( for n being Nat holds IT2 . (n + 1) = AP:Step (IT2 . n) ) implies IT1 = IT2 )
assume that
A9: ( IT1 . 0 = source .--> 1 & ( for n being Nat holds IT1 . (n + 1) = AP:Step (IT1 . n) ) ) and
A10: ( IT2 . 0 = source .--> 1 & ( for n being Nat holds IT2 . (n + 1) = AP:Step (IT2 . n) ) ) ; :: thesis: IT1 = IT2
defpred S₁[ Nat] means IT1 . $1 = IT2 . $1;
A11: S₁[ 0 ] by A9, A10;

now

let n be Nat; :: thesis: ( S₁[n] implies S₁[n + 1] )
assume S₁[n] ; :: thesis: S₁[n + 1]
then IT1 . (n + 1) = AP:Step (IT2 . n) by A9
.= IT2 . (n + 1) by A10 ;
hence S₁[n + 1] ; :: thesis: verum

end;

then A12: for n being Nat st S₁[n] holds
S₁[n + 1] ;
A12a: for n being Nat holds S₁[n] from NAT_1:sch 2(A11, A12);
for n being set st n in NAT holds
IT1 . n = IT2 . n by A12a;
hence IT1 = IT2 by PBOOLE:3; :: thesis: verum