hereby :: thesis: ( ( for x being Element of dom GF holds GF . x is connected ) implies GF is connected )

assume A3:
for x being Element of dom GF holds GF . x is connected
; :: thesis: GF is connected assume A1:
GF is connected
; :: thesis: for x being Element of dom GF holds GF . x is connected

let x be Element of dom GF; :: thesis: GF . x is connected

consider G being _Graph such that

A2: ( GF . x = G & G is connected ) by A1;

thus GF . x is connected by A2; :: thesis: verum

end;let x be Element of dom GF; :: thesis: GF . x is connected

consider G being _Graph such that

A2: ( GF . x = G & G is connected ) by A1;

thus GF . x is connected by A2; :: thesis: verum

let x be object ; :: according to GLIB_002:def 12 :: thesis: ( x in dom GF implies ex G being _Graph st

( GF . x = G & G is connected ) )

assume x in dom GF ; :: thesis: ex G being _Graph st

( GF . x = G & G is connected )

then reconsider y = x as Element of dom GF ;

take GF . y ; :: thesis: ( GF . x = GF . y & GF . y is connected )

thus ( GF . x = GF . y & GF . y is connected ) by A3; :: thesis: verum