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

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

hence
f1 = f2
by A4, A6, FUNCT_1:2; :: thesis: verumf1 . 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;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