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
A4: dom f1 = Funcs (P,NAT) and
A5: for m being marking of P holds f1 . m = fire (t,m) and
A6: dom f2 = Funcs (P,NAT) and
A7: for m being marking of P holds f2 . m = fire (t,m) ; :: thesis: f1 = f2

now :: thesis: for x being object st x in dom f1 holds
f1 . x = f2 . x

let x be object ; :: 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 A4, FUNCT_2:66;
thus f1 . x = fire (t,m) by A5
.= f2 . x by A7 ; :: thesis: verum

end;

hence f1 = f2 by A4, A6, FUNCT_1:2; :: thesis: verum