let X be set ; :: thesis: for R being Relation st R well_orders X holds

for Y being set st Y c= X & Y <> {} holds

ex a being object st

( a in Y & ( for b being object st b in Y holds

[a,b] in R ) )

let R be Relation; :: thesis: ( R well_orders X implies for Y being set st Y c= X & Y <> {} holds

ex a being object st

( a in Y & ( for b being object st b in Y holds

[a,b] in R ) ) )

assume A1: R well_orders X ; :: thesis: for Y being set st Y c= X & Y <> {} holds

ex a being object st

( a in Y & ( for b being object st b in Y holds

[a,b] in R ) )

then A2: R is_reflexive_in X ;

A3: R is_connected_in X by A1;

let Y be set ; :: thesis: ( Y c= X & Y <> {} implies ex a being object st

( a in Y & ( for b being object st b in Y holds

[a,b] in R ) ) )

assume that

A4: Y c= X and

A5: Y <> {} ; :: thesis: ex a being object st

( a in Y & ( for b being object st b in Y holds

[a,b] in R ) )

R is_well_founded_in X by A1;

then consider a being object such that

A6: a in Y and

A7: R -Seg a misses Y by A4, A5;

take a ; :: thesis: ( a in Y & ( for b being object st b in Y holds

[a,b] in R ) )

thus a in Y by A6; :: thesis: for b being object st b in Y holds

[a,b] in R

let b be object ; :: thesis: ( b in Y implies [a,b] in R )

assume A8: b in Y ; :: thesis: [a,b] in R

then not b in R -Seg a by A7, XBOOLE_0:3;

then ( a = b or not [b,a] in R ) by Th1;

then ( a <> b implies [a,b] in R ) by A3, A4, A6, A8, RELAT_2:def 6;

hence [a,b] in R by A2, A4, A6, RELAT_2:def 1; :: thesis: verum

for Y being set st Y c= X & Y <> {} holds

ex a being object st

( a in Y & ( for b being object st b in Y holds

[a,b] in R ) )

let R be Relation; :: thesis: ( R well_orders X implies for Y being set st Y c= X & Y <> {} holds

ex a being object st

( a in Y & ( for b being object st b in Y holds

[a,b] in R ) ) )

assume A1: R well_orders X ; :: thesis: for Y being set st Y c= X & Y <> {} holds

ex a being object st

( a in Y & ( for b being object st b in Y holds

[a,b] in R ) )

then A2: R is_reflexive_in X ;

A3: R is_connected_in X by A1;

let Y be set ; :: thesis: ( Y c= X & Y <> {} implies ex a being object st

( a in Y & ( for b being object st b in Y holds

[a,b] in R ) ) )

assume that

A4: Y c= X and

A5: Y <> {} ; :: thesis: ex a being object st

( a in Y & ( for b being object st b in Y holds

[a,b] in R ) )

R is_well_founded_in X by A1;

then consider a being object such that

A6: a in Y and

A7: R -Seg a misses Y by A4, A5;

take a ; :: thesis: ( a in Y & ( for b being object st b in Y holds

[a,b] in R ) )

thus a in Y by A6; :: thesis: for b being object st b in Y holds

[a,b] in R

let b be object ; :: thesis: ( b in Y implies [a,b] in R )

assume A8: b in Y ; :: thesis: [a,b] in R

then not b in R -Seg a by A7, XBOOLE_0:3;

then ( a = b or not [b,a] in R ) by Th1;

then ( a <> b implies [a,b] in R ) by A3, A4, A6, A8, RELAT_2:def 6;

hence [a,b] in R by A2, A4, A6, RELAT_2:def 1; :: thesis: verum