let R be Relation; :: thesis: for x being set st x is_minimal_in R & R is connected holds
x is_inferior_of R
let x be set ; :: thesis: ( x is_minimal_in R & R is connected implies x is_inferior_of R )
assume that
A1: x is_minimal_in R and
A2: R is connected ; :: thesis: x is_inferior_of R
thus A3: x in field R by A1; :: according to ORDERS_1:def 15 :: thesis: for y being set st y in field R & y <> x holds
[x,y] in R
let y be set ; :: thesis: ( y in field R & y <> x implies [x,y] in R )
assume that
A4: y in field R and
A5: y <> x ; :: thesis: [x,y] in R
R is_connected_in field R by A2;
then ( [x,y] in R or [y,x] in R ) by A3, A4, A5;
hence [x,y] in R by A1, A4; :: thesis: verum