A1: { s where s is Element of T : not t is_a_proper_prefix_of s } c= T

proof

let x be object ; :: according to TARSKI:def 3 :: thesis: ( not x in { s where s is Element of T : not t is_a_proper_prefix_of s } or x in T )
assume x in { s where s is Element of T : not t is_a_proper_prefix_of s } ; :: thesis: x in T
then ex s being Element of T st
( x = s & not t is_a_proper_prefix_of s ) ;
hence x in T ; :: thesis: verum

end;

T1, { (t ^ s) where s is Element of T1 : verum } are_equipotent

proof

defpred S₁[ object , object ] means ex q being FinSequence of NAT st
( T = q & t = t ^ q );
A2: for x being object st x in T1 holds
ex y being object st S₁[x,y]

proof

let x be object ; :: thesis: ( x in T1 implies ex y being object st S₁[x,y] )
assume x in T1 ; :: thesis: ex y being object st S₁[x,y]
then reconsider q = x as FinSequence of NAT by Th18;
t ^ q = t ^ q ;
hence ex y being object st S₁[x,y] ; :: thesis: verum

end;

consider f being Function such that
A3: ( dom f = T1 & ( for x being object st x in T1 holds
S₁[x,f . x] ) ) from CLASSES1:sch 1(A2);
take f ; :: according to WELLORD2:def 4 :: thesis: ( f is one-to-one & dom f = T1 & rng f = { (t ^ s) where s is Element of T1 : verum } )
thus f is one-to-one :: thesis: ( dom f = T1 & rng f = { (t ^ s) where s is Element of T1 : verum } )

proof

let x, y be object ; :: according to FUNCT_1:def 4 :: thesis: ( not x in dom f or not y in dom f or not f . x = f . y or x = y )
assume that
A4: ( x in dom f & y in dom f ) and
A5: f . x = f . y ; :: thesis: x = y
( ex q being FinSequence of NAT st
( x = q & f . x = t ^ q ) & ex r being FinSequence of NAT st
( y = r & f . y = t ^ r ) ) by A3, A4;
hence x = y by A5, FINSEQ_1:33; :: thesis: verum

end;

thus dom f = T1 by A3; :: thesis: rng f = { (t ^ s) where s is Element of T1 : verum }
thus rng f c= { (t ^ s) where s is Element of T1 : verum } :: according to XBOOLE_0:def 10 :: thesis: { (t ^ s) where s is Element of T1 : verum } is_a_prefix_of rng f

proof

let x be object ; :: according to TARSKI:def 3 :: thesis: ( not x in rng f or x in { (t ^ s) where s is Element of T1 : verum } )
assume x in rng f ; :: thesis: x in { (t ^ s) where s is Element of T1 : verum }
then consider y being object such that
A6: y in dom f and
A7: x = f . y by FUNCT_1:def 3;
consider q being FinSequence of NAT such that
A8: y = q and
A9: f . y = t ^ q by A3, A6;
reconsider q = q as Element of T1 by A3, A6, A8;
x = t ^ q by A7, A9;
hence x in { (t ^ s) where s is Element of T1 : verum } ; :: thesis: verum

end;

let x be object ; :: according to TARSKI:def 3 :: thesis: ( not x in { (t ^ s) where s is Element of T1 : verum } or x in rng f )
assume x in { (t ^ s) where s is Element of T1 : verum } ; :: thesis: x in rng f
then consider s being Element of T1 such that
A10: x = t ^ s ;
S₁[s,f . s] by A3;
hence x in rng f by A3, A10, FUNCT_1:def 3; :: thesis: verum

end;

then { (t ^ s) where s is Element of T1 : verum } is finite by CARD_1:38;
then { v where v is Element of T : not t is_a_proper_prefix_of v } \/ { (t ^ s) where s is Element of T1 : verum } is finite by A1;
hence T with-replacement (t,T1) is finite by Th31; :: thesis: verum