set wfp = well_founded-Part R;
set r = the InternalRel of R;
set IT = { S where S is Subset of R : ( S is well_founded & S is lower ) } ;
A1: well_founded-Part R = union { S where S is Subset of R : ( S is well_founded & S is lower ) } by Def4;

hereby :: according to WELLFND1:def 1 :: thesis: well_founded-Part R is well_founded

let x, y be object ; :: thesis: ( x in well_founded-Part R & [y,x] in the InternalRel of R implies y in well_founded-Part R )
assume that
A2: x in well_founded-Part R and
A3: [y,x] in the InternalRel of R ; :: thesis: y in well_founded-Part R
consider Y being set such that
A4: x in Y and
A5: Y in { S where S is Subset of R : ( S is well_founded & S is lower ) } by A1, A2, TARSKI:def 4;
consider S being Subset of R such that
A6: Y = S and
A7: ( S is well_founded & S is lower ) by A5;
y in S by A3, A4, A6, A7;
hence y in well_founded-Part R by A1, A5, A6, TARSKI:def 4; :: thesis: verum

end;

let Y be set ; :: according to WELLORD1:def 3,WELLFND1:def 3 :: thesis: ( not Y c= well_founded-Part R or Y = {} or ex b₁ being object st
( b₁ in Y & the InternalRel of R -Seg b₁ misses Y ) )
assume that
A8: Y c= well_founded-Part R and
A9: Y <> {} ; :: thesis: ex b₁ being object st
( b₁ in Y & the InternalRel of R -Seg b₁ misses Y )
consider y being object such that
A10: y in Y by A9, XBOOLE_0:def 1;
consider YY being set such that
A11: y in YY and
A12: YY in { S where S is Subset of R : ( S is well_founded & S is lower ) } by A1, A8, A10, TARSKI:def 4;
consider S being Subset of R such that
A13: YY = S and
A14: ( S is well_founded & S is lower ) by A12;
set YS = Y /\ S;
A15: the InternalRel of R is_well_founded_in S by A14;
( Y /\ S c= S & Y /\ S <> {} ) by A10, A11, A13, XBOOLE_0:def 4;
then consider a being object such that
A16: a in Y /\ S and
A17: the InternalRel of R -Seg a misses Y /\ S by A15;
A18: a in Y by A16, XBOOLE_0:def 4;
a in S by A16, XBOOLE_0:def 4;
then A19: ( the InternalRel of R -Seg a) /\ Y = (( the InternalRel of R -Seg a) /\ S) /\ Y by A14, Th4, XBOOLE_1:28
.= ( the InternalRel of R -Seg a) /\ (Y /\ S) by XBOOLE_1:16 ;
( the InternalRel of R -Seg a) /\ (Y /\ S) = {} by A17;
then the InternalRel of R -Seg a misses Y by A19;
hence ex b₁ being object st
( b₁ in Y & the InternalRel of R -Seg b₁ misses Y ) by A18; :: thesis: verum