let IT1, IT2 be MCS:LabelingSeq of G; :: thesis: ( IT1 . 0 = MCS:Init G & ( for n being Nat holds IT1 . (n + 1) = MCS:Step (IT1 . n) ) & IT2 . 0 = MCS:Init G & ( for n being Nat holds IT2 . (n + 1) = MCS:Step (IT2 . n) ) implies IT1 = IT2 )
assume that
A11: ( IT1 . 0 = MCS:Init G & ( for n being Nat holds IT1 . (n + 1) = MCS:Step (IT1 . n) ) ) and
A12: ( IT2 . 0 = MCS:Init G & ( for n being Nat holds IT2 . (n + 1) = MCS:Step (IT2 . n) ) ) ; :: thesis: IT1 = IT2
defpred S₁[ Nat] means IT1 . $1 = IT2 . $1;
A13: S₁[ 0 ] by A11, A12;

now

let n be Nat; :: thesis: ( S₁[n] implies S₁[n + 1] )
assume S₁[n] ; :: thesis: S₁[n + 1]
then IT1 . (n + 1) = MCS:Step (IT2 . n) by A11
.= IT2 . (n + 1) by A12 ;
hence S₁[n + 1] ; :: thesis: verum

end;

then A14: for n being Element of NAT st S₁[n] holds
S₁[n + 1] ;
for n being Element of NAT holds S₁[n] from NAT_1:sch 1(A13, A14);
then for n being set st n in NAT holds
IT1 . n = IT2 . n ;
hence IT1 = IT2 by PBOOLE:3; :: thesis: verum