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

assume A1: GF is acyclic ; :: thesis: for x being Element of dom GF holds GF . x is acyclic
let x be Element of dom GF; :: thesis: GF . x is acyclic
consider G being _Graph such that
A2: ( GF . x = G & G is acyclic ) by A1;
thus GF . x is acyclic by A2; :: thesis: verum

end;

assume A3: for x being Element of dom GF holds GF . x is acyclic ; :: thesis: GF is acyclic
let x be object ; :: according to GLIB_002:def 13 :: thesis: ( x in dom GF implies ex G being _Graph st
( GF . x = G & G is acyclic ) )
assume x in dom GF ; :: thesis: ex G being _Graph st
( GF . x = G & G is acyclic )
then reconsider y = x as Element of dom GF ;
take GF . y ; :: thesis: ( GF . x = GF . y & GF . y is acyclic )
thus ( GF . x = GF . y & GF . y is acyclic ) by A3; :: thesis: verum