let n be Nat; :: thesis: ( omega *` (card n) c= omega & (card n) *` omega c= omega )
defpred S₂[ Nat] means omega *` (card $1) c= omega ;
omega *` (card 0 ) = 0 by CARD_2:32;
then A1: S₂[ 0 ] ;
A2: for k being Nat st S₂[k] holds
S₂[k + 1]

proof

let k be Nat; :: thesis: ( S₂[k] implies S₂[k + 1] )
assume A3: S₂[k] ; :: thesis: S₂[k + 1]
card (k + 1) = k + 1 by CARD_1:def 5
.= succ k by NAT_1:39 ;
then card (k + 1) = card (succ k) ;
then omega *` (card (k + 1)) = card ((succ k) *^ omega ) by CARD_1:84, CARD_2:25
.= card ((k *^ omega ) +^ omega ) by ORDINAL2:53
.= (card (k *^ omega )) +` omega by CARD_1:84, CARD_2:24
.= (omega *` (card k)) +` omega by CARD_1:84, CARD_2:25
.= omega by A3, Th118 ;
hence S₂[k + 1] ; :: thesis: verum

end;

A4: for k being Nat holds S₂[k] from NAT_1:sch 2(A1, A2);
hence omega *` (card n) c= omega ; :: thesis: (card n) *` omega c= omega
thus (card n) *` omega c= omega by A4; :: thesis: verum