let G be _Graph; :: thesis: ( G is loopless iff for v, e being object holds not e DSJoins {v},{v},G )

for v, e being object holds not e SJoins {v},{v},G

hereby :: thesis: ( ( for v, e being object holds not e DSJoins {v},{v},G ) implies G is loopless )

assume A3:
for v, e being object holds not e DSJoins {v},{v},G
; :: thesis: G is loopless assume A1:
G is loopless
; :: thesis: for v, e being object holds not e DSJoins {v},{v},G

let v be object ; :: thesis: for e being object holds not e DSJoins {v},{v},G

given e being object such that A2: e DSJoins {v},{v},G ; :: thesis: contradiction

e SJoins {v},{v},G by A2, Th11;

hence contradiction by A1, Th18; :: thesis: verum

end;let v be object ; :: thesis: for e being object holds not e DSJoins {v},{v},G

given e being object such that A2: e DSJoins {v},{v},G ; :: thesis: contradiction

e SJoins {v},{v},G by A2, Th11;

hence contradiction by A1, Th18; :: thesis: verum

for v, e being object holds not e SJoins {v},{v},G

proof

hence
G is loopless
by Th18; :: thesis: verum
let v be object ; :: thesis: for e being object holds not e SJoins {v},{v},G

given e being object such that A4: e SJoins {v},{v},G ; :: thesis: contradiction

e DSJoins {v},{v},G by A4, Th11;

hence contradiction by A3; :: thesis: verum

end;given e being object such that A4: e SJoins {v},{v},G ; :: thesis: contradiction

e DSJoins {v},{v},G by A4, Th11;

hence contradiction by A3; :: thesis: verum