let R be RelStr ; :: thesis: ( R \~ is well_founded & R is antisymmetric implies R is well_founded )

assume that

A1: R \~ is well_founded and

A2: R is antisymmetric ; :: thesis: R is well_founded

set IR = the InternalRel of R;

set CR = the carrier of R;

set IR9 = the InternalRel of (R \~);

A3: the InternalRel of R is_antisymmetric_in the carrier of R by A2;

A4: the InternalRel of (R \~) is_well_founded_in the carrier of R by A1, WELLFND1:def 2;

hence R is well_founded by WELLFND1:def 2; :: thesis: verum

assume that

A1: R \~ is well_founded and

A2: R is antisymmetric ; :: thesis: R is well_founded

set IR = the InternalRel of R;

set CR = the carrier of R;

set IR9 = the InternalRel of (R \~);

A3: the InternalRel of R is_antisymmetric_in the carrier of R by A2;

A4: the InternalRel of (R \~) is_well_founded_in the carrier of R by A1, WELLFND1:def 2;

now :: thesis: for Y being set st Y c= the carrier of R & Y <> {} holds

ex a being object st

( a in Y & the InternalRel of R -Seg a misses Y )

then
the InternalRel of R is_well_founded_in the carrier of R
by WELLORD1:def 3;ex a being object st

( a in Y & the InternalRel of R -Seg a misses Y )

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

( a in Y & the InternalRel of R -Seg a misses Y ) )

assume that

A5: Y c= the carrier of R and

A6: Y <> {} ; :: thesis: ex a being object st

( a in Y & the InternalRel of R -Seg a misses Y )

consider a being object such that

A7: a in Y and

A8: the InternalRel of (R \~) -Seg a misses Y by A4, A5, A6, WELLORD1:def 3;

A9: ( the InternalRel of (R \~) -Seg a) /\ Y = {} by A8, XBOOLE_0:def 7;

take a = a; :: thesis: ( a in Y & the InternalRel of R -Seg a misses Y )

thus a in Y by A7; :: thesis: the InternalRel of R -Seg a misses Y

end;( a in Y & the InternalRel of R -Seg a misses Y ) )

assume that

A5: Y c= the carrier of R and

A6: Y <> {} ; :: thesis: ex a being object st

( a in Y & the InternalRel of R -Seg a misses Y )

consider a being object such that

A7: a in Y and

A8: the InternalRel of (R \~) -Seg a misses Y by A4, A5, A6, WELLORD1:def 3;

A9: ( the InternalRel of (R \~) -Seg a) /\ Y = {} by A8, XBOOLE_0:def 7;

take a = a; :: thesis: ( a in Y & the InternalRel of R -Seg a misses Y )

thus a in Y by A7; :: thesis: the InternalRel of R -Seg a misses Y

now :: thesis: not ( the InternalRel of R -Seg a) /\ Y <> {}

hence
the InternalRel of R -Seg a misses Y
by XBOOLE_0:def 7; :: thesis: verumassume
( the InternalRel of R -Seg a) /\ Y <> {}
; :: thesis: contradiction

then consider z being object such that

A10: z in ( the InternalRel of R -Seg a) /\ Y by XBOOLE_0:def 1;

A11: z in the InternalRel of R -Seg a by A10, XBOOLE_0:def 4;

A12: z in Y by A10, XBOOLE_0:def 4;

A13: z <> a by A11, WELLORD1:1;

A14: [z,a] in the InternalRel of R by A11, WELLORD1:1;

then not [a,z] in the InternalRel of R by A3, A5, A7, A12, A13;

then not [z,a] in the InternalRel of R ~ by RELAT_1:def 7;

then [z,a] in the InternalRel of R \ ( the InternalRel of R ~) by A14, XBOOLE_0:def 5;

then z in the InternalRel of (R \~) -Seg a by A13, WELLORD1:1;

hence contradiction by A9, A12, XBOOLE_0:def 4; :: thesis: verum

end;then consider z being object such that

A10: z in ( the InternalRel of R -Seg a) /\ Y by XBOOLE_0:def 1;

A11: z in the InternalRel of R -Seg a by A10, XBOOLE_0:def 4;

A12: z in Y by A10, XBOOLE_0:def 4;

A13: z <> a by A11, WELLORD1:1;

A14: [z,a] in the InternalRel of R by A11, WELLORD1:1;

then not [a,z] in the InternalRel of R by A3, A5, A7, A12, A13;

then not [z,a] in the InternalRel of R ~ by RELAT_1:def 7;

then [z,a] in the InternalRel of R \ ( the InternalRel of R ~) by A14, XBOOLE_0:def 5;

then z in the InternalRel of (R \~) -Seg a by A13, WELLORD1:1;

hence contradiction by A9, A12, XBOOLE_0:def 4; :: thesis: verum

hence R is well_founded by WELLFND1:def 2; :: thesis: verum