let f1, f2 be Function; :: thesis: ( dom f1 = Funcs P,NAT & ( for m being marking of P holds f1 . m = fire t,m ) & dom f2 = Funcs P,NAT & ( for m being marking of P holds f2 . m = fire t,m ) implies f1 = f2 )
assume that
A5: dom f1 = Funcs P,NAT and
A6: for m being marking of P holds f1 . m = fire t,m and
A7: dom f2 = Funcs P,NAT and
A8: for m being marking of P holds f2 . m = fire t,m ; :: thesis: f1 = f2

now

let x be set ; :: thesis: ( x in dom f1 implies f1 . x = f2 . x )
assume x in dom f1 ; :: thesis: f1 . x = f2 . x
then reconsider m = x as marking of P by A5, FUNCT_2:121;
thus f1 . x = fire t,m by A6
.= f2 . x by A8 ; :: thesis: verum

end;

hence f1 = f2 by A5, A7, FUNCT_1:9; :: thesis: verum