let X be non empty set ; :: thesis: for S being SigmaField of X
for n being Nat
for F being Relation st F is Finite_Sep_Sequence of S holds
F | (Seg n) is Finite_Sep_Sequence of S
let S be SigmaField of X; :: thesis: for n being Nat
for F being Relation st F is Finite_Sep_Sequence of S holds
F | (Seg n) is Finite_Sep_Sequence of S
let n be Nat; :: thesis: for F being Relation st F is Finite_Sep_Sequence of S holds
F | (Seg n) is Finite_Sep_Sequence of S
let F be Relation; :: thesis: ( F is Finite_Sep_Sequence of S implies F | (Seg n) is Finite_Sep_Sequence of S )
assume A1: F is Finite_Sep_Sequence of S ; :: thesis: F | (Seg n) is Finite_Sep_Sequence of S
then reconsider G = F | (Seg n) as FinSequence of S by FINSEQ_1:23;
reconsider F = F as FinSequence of S by A1;
for k, m being set st k <> m holds
G . k misses G . m

proof

let k, m be set ; :: thesis: ( k <> m implies G . k misses G . m )
assume A3: k <> m ; :: thesis: G . k misses G . m

per cases ( ( k in dom G & m in dom G ) or not k in dom G or not m in dom G ) ;

suppose ( k in dom G & m in dom G ) ; :: thesis: G . k misses G . m

then ( G . k = F . k & G . m = F . m ) by FUNCT_1:70;
hence G . k misses G . m by A1, A3, PROB_2:def 3; :: thesis: verum

end;

suppose ( not k in dom G or not m in dom G ) ; :: thesis: G . k misses G . m

then ( G . k = {} or G . m = {} ) by FUNCT_1:def 4;
hence G . k misses G . m by XBOOLE_1:65; :: thesis: verum

end;

end;

end;

hence F | (Seg n) is Finite_Sep_Sequence of S by PROB_2:def 3; :: thesis: verum