let X be set ; :: thesis: for x, y being object
for R being Relation of X st X c= field R & R reduces x,y & x in X holds
[x,y] in R [*]
let x, y be object ; :: thesis: for R being Relation of X st X c= field R & R reduces x,y & x in X holds
[x,y] in R [*]
let R be Relation of X; :: thesis: ( X c= field R & R reduces x,y & x in X implies [x,y] in R [*] )
assume A1: X c= field R ; :: thesis: ( not R reduces x,y or not x in X or [x,y] in R [*] )
assume that
A2: R reduces x,y and
A3: x in X ; :: thesis: [x,y] in R [*]

per cases ( [x,y] in R [*] or x = y ) by A2, REWRITE1:20;

suppose [x,y] in R [*] ; :: thesis: [x,y] in R [*]

hence [x,y] in R [*] ; :: thesis: verum

end;

suppose A4: x = y ; :: thesis: [x,y] in R [*]

R [*] is_reflexive_in X by A1, Th5;
hence [x,y] in R [*] by A3, A4, RELAT_2:def 1; :: thesis: verum

end;

end;