defpred S₁[ set ] means ( ex m being non empty Element of NAT st
( m <= n & $1 = PFArt m,k ) or $1 = PFCrt n,k );
A17: for X1, X2 being Element of Fin (PFuncs NAT ,{k}) st ( for x being set holds
( x in X1 iff S₁[x] ) ) & ( for x being set holds
( x in X2 iff S₁[x] ) ) holds
X1 = X2 from LMOD_7:sch 1();
let X1, X2 be Element of Fin (PFuncs NAT ,{k}); :: thesis: ( ( for x being set holds
( x in X1 iff ( ex m being non empty Element of NAT st
( m <= n & x = PFArt m,k ) or x = PFCrt n,k ) ) ) & ( for x being set holds
( x in X2 iff ( ex m being non empty Element of NAT st
( m <= n & x = PFArt m,k ) or x = PFCrt n,k ) ) ) implies X1 = X2 )
assume ( ( for x being set holds
( x in X1 iff S₁[x] ) ) & ( for x being set holds
( x in X2 iff S₁[x] ) ) ) ; :: thesis: X1 = X2
hence X1 = X2 by A17; :: thesis: verum