let S be non empty non void ManySortedSign ; :: thesis: for s being SortSymbol of S
for X being non-empty ManySortedSet of the carrier of S
for x being Element of X . s
for t, t1 being Element of (Free (S,X))
for xi being Element of dom t st t . xi = [x,s] holds
dom t c= dom (t with-replacement (xi,t1))
let s be SortSymbol of S; :: thesis: for X being non-empty ManySortedSet of the carrier of S
for x being Element of X . s
for t, t1 being Element of (Free (S,X))
for xi being Element of dom t st t . xi = [x,s] holds
dom t c= dom (t with-replacement (xi,t1))
let X be non-empty ManySortedSet of the carrier of S; :: thesis: for x being Element of X . s
for t, t1 being Element of (Free (S,X))
for xi being Element of dom t st t . xi = [x,s] holds
dom t c= dom (t with-replacement (xi,t1))
let x be Element of X . s; :: thesis: for t, t1 being Element of (Free (S,X))
for xi being Element of dom t st t . xi = [x,s] holds
dom t c= dom (t with-replacement (xi,t1))
let t, t1 be Element of (Free (S,X)); :: thesis: for xi being Element of dom t st t . xi = [x,s] holds
dom t c= dom (t with-replacement (xi,t1))
let xi be Element of dom t; :: thesis: ( t . xi = [x,s] implies dom t c= dom (t with-replacement (xi,t1)) )
assume Z0: t . xi = [x,s] ; :: thesis: dom t c= dom (t with-replacement (xi,t1))
let a be object ; :: according to TARSKI:def 3 :: thesis: ( not a in dom t or a in dom (t with-replacement (xi,t1)) )
assume a in dom t ; :: thesis: a in dom (t with-replacement (xi,t1))
then reconsider q = a as Element of dom t ;
xi in Leaves (dom t) by Z0, Lem13;
then not xi c< q by TREES_1:def 5;
then q in (dom t) with-replacement (xi,(dom t1)) by TREES_1:def 9;
hence a in dom (t with-replacement (xi,t1)) by TREES_2:def 11; :: thesis: verum