let S1, S2 be SetSequence of ExtREAL; :: thesis: ( S1 . 0 = ].(b - 1),+infty.] & ( for n being Nat holds S1 . (n + 1) = ].(b - (1 / (n + 1))),+infty.] ) & S2 . 0 = ].(b - 1),+infty.] & ( for n being Nat holds S2 . (n + 1) = ].(b - (1 / (n + 1))),+infty.] ) implies S1 = S2 )
assume that
A4: ( S1 . 0 = ].(b - 1),+infty.] & ( for n being Nat holds S1 . (n + 1) = ].(b - (1 / (n + 1))),+infty.] ) ) and
A5: ( S2 . 0 = ].(b - 1),+infty.] & ( for n being Nat holds S2 . (n + 1) = ].(b - (1 / (n + 1))),+infty.] ) ) ; :: thesis: S1 = S2
defpred S1[ object ] means S1 . \$1 = S2 . \$1;
for n being object st n in NAT holds
S1[n]
proof
let n be object ; :: thesis: ( n in NAT implies S1[n] )
assume n in NAT ; :: thesis: S1[n]
then reconsider n = n as Element of NAT ;
A6: S1[ 0 ] by A4, A5;
A7: for k being Nat st S1[k] holds
S1[k + 1]
proof
let k be Nat; :: thesis: ( S1[k] implies S1[k + 1] )
assume S1 . k = S2 . k ; :: thesis: S1[k + 1]
thus S1 . (k + 1) = ].(b - (1 / (k + 1))),+infty.] by A4
.= S2 . (k + 1) by A5 ; :: thesis: verum
end;
for k being Nat holds S1[k] from NAT_1:sch 2(A6, A7);
then S1 . n = S2 . n ;
hence S1[n] ; :: thesis: verum
end;
hence S1 = S2 ; :: thesis: verum