let X be non empty set ; :: thesis:
let S be SigmaField of X; :: thesis:
let F be Function; :: thesis:
defpred S₁[ object , object ] means ( ( $1 in dom F implies F . $1 = $2 ) & ( not $1 in dom F implies $2 = {} ) );
assume A1: F is Finite_Sep_Sequence of S ; :: thesis:
A2: for x1 being object st x1 in NAT holds
ex y1 being object st
( y1 in S & S₁[x1,y1] )

proof

let x1 be object ; :: thesis:
assume x1 in NAT ; :: thesis:
then reconsider x1 = x1 as Element of NAT ;

per cases ;

suppose A3: x1 in dom F ; :: thesis:

then A4: F . x1 in rng F by FUNCT_1:def 3;
A5: rng F c= S by A1, FINSEQ_1:def 4;
take F . x1 ; :: thesis:
thus ( F . x1 in S & S₁[x1,F . x1] ) by A3, A4, A5; :: thesis:

end;

suppose A6: not x1 in dom F ; :: thesis:

take {} ; :: thesis:
thus ( {} in S & S₁[x1, {} ] ) by A6, PROB_1:4; :: thesis:

end;

end;

end;

consider G being sequence of S such that
A7: for x1 being object st x1 in NAT holds
S₁[x1,G . x1] from FUNCT_2:sch 1(A2);
for n, m being object st n <> m holds
G . n misses G . m

proof

let n, m be object ; :: thesis:
assume A8: n <> m ; :: thesis:

per cases ;

suppose A9: ( n in NAT & m in NAT ) ; :: thesis:

now :: thesis:

per cases ;

suppose ( n in dom F & m in dom F ) ; :: thesis:

then ( G . n = F . n & G . m = F . m ) by A7, A9;
hence G . n misses G . m by A1, A8, PROB_2:def 2; :: thesis:

end;

suppose A10: ( not n in dom F or not m in dom F ) ; :: thesis:

now :: thesis:

per cases by A10;

suppose not n in dom F ; :: thesis:

then G . n = {} by A7, A9;
hence G . n misses G . m ; :: thesis:

end;

suppose not m in dom F ; :: thesis:

then G . m = {} by A7, A9;
hence G . n misses G . m ; :: thesis:

end;

end;

end;

hence G . n misses G . m ; :: thesis:

end;

end;

end;

hence G . n misses G . m ; :: thesis:

end;

suppose ( not n in NAT or not m in NAT ) ; :: thesis:

then ( not n in dom G or not m in dom G ) ;
then ( G . n = {} or G . m = {} ) by FUNCT_1:def 2;
hence G . n misses G . m ; :: thesis:

end;

end;

end;

then reconsider G = G as Sep_Sequence of S by PROB_2:def 2;
take G ; :: thesis:
for x1 being object st x1 in union (rng F) holds
x1 in union (rng G)

proof

let x1 be object ; :: thesis:
assume x1 in union (rng F) ; :: thesis:
then consider Y being set such that
A11: x1 in Y and
A12: Y in rng F by TARSKI:def 4;
consider k being object such that
A13: k in dom F and
A14: Y = F . k by A12, FUNCT_1:def 3;
dom F c= NAT by A1, RELAT_1:def 18;
then reconsider k = k as Element of NAT by A13;
A15: F . k = G . k by A7, A13;
G . k in rng G by FUNCT_2:4;
hence x1 in union (rng G) by A11, A14, A15, TARSKI:def 4; :: thesis:

end;

then A16: union (rng F) c= union (rng G) ;
for x1 being object st x1 in union (rng G) holds
x1 in union (rng F)

proof

let x1 be object ; :: thesis:
assume x1 in union (rng G) ; :: thesis:
then consider Y being set such that
A17: x1 in Y and
A18: Y in rng G by TARSKI:def 4;
consider k being object such that
A19: k in dom G and
A20: Y = G . k by A18, FUNCT_1:def 3;
reconsider k = k as Element of NAT by A19;
A21: k in dom F by A7, A17, A20;
A22: F . k = G . k by A7, A17, A20;
F . k in rng F by A21, FUNCT_1:def 3;
hence x1 in union (rng F) by A17, A20, A22, TARSKI:def 4; :: thesis:

end;

then union (rng G) c= union (rng F) ;
hence union (rng F) = union (rng G) by A16; :: thesis:

hereby :: thesis:

let n be Nat; :: thesis:
n in NAT by ORDINAL1:def 12;
hence ( n in dom F implies F . n = G . n ) by A7; :: thesis:

end;

let m be Nat; :: thesis:
m in NAT by ORDINAL1:def 12;
hence ( not m in dom F implies G . m = {} ) by A7; :: thesis: