defpred S₁[ set ] means ex B' being Element of Fin F₁() st
( B' = $1 & P₁[B'] );
let B be Element of Fin F₁(); :: thesis: P₁[B]
A3: for x, A being set st x in B & A c= B & S₁[A] holds
S₁[A \/ {x}]

proof

let x, A be set ; :: thesis: ( x in B & A c= B & S₁[A] implies S₁[A \/ {x}] )
assume that
A4: x in B and
A5: A c= B and
A6: ex B' being Element of Fin F₁() st
( B' = A & P₁[B'] ) ; :: thesis: S₁[A \/ {x}]
reconsider x' = x as Element of F₁() by A4, Th14;
reconsider A' = A as Element of Fin F₁() by A5, Th16;
take A' \/ {.x'.} ; :: thesis: ( A' \/ {.x'.} = A \/ {x} & P₁[A' \/ {.x'.}] )
thus A' \/ {.x'.} = A \/ {x} ; :: thesis: P₁[A' \/ {.x'.}]
thus P₁[A' \/ {x'}] by A2, A6, ZFMISC_1:46; :: thesis: verum

end;

A7: B is finite ;
A8: S₁[ {} ] by A1;
S₁[B] from FINSET_1:sch 2(A7, A8, A3);
hence P₁[B] ; :: thesis: verum