let m be non empty Element of NAT ; :: thesis: for x, y being Element of REAL
for i being Element of NAT st 1 <= i & i <= m holds
Replace (0* m),i,(x + y) = (Replace (0* m),i,x) + (Replace (0* m),i,y)
let x, y be Element of REAL ; :: thesis: for i being Element of NAT st 1 <= i & i <= m holds
Replace (0* m),i,(x + y) = (Replace (0* m),i,x) + (Replace (0* m),i,y)
let i be Element of NAT ; :: thesis: ( 1 <= i & i <= m implies Replace (0* m),i,(x + y) = (Replace (0* m),i,x) + (Replace (0* m),i,y) )
assume A1: ( 1 <= i & i <= m ) ; :: thesis: Replace (0* m),i,(x + y) = (Replace (0* m),i,x) + (Replace (0* m),i,y)
A2: ( len (Replace (0* m),i,(x + y)) = m & len (Replace (0* m),i,x) = m & len (Replace (0* m),i,y) = m ) by Lm6;
then A3: len ((Replace (0* m),i,x) + (Replace (0* m),i,y)) = len (Replace (0* m),i,(x + y)) by RVSUM_1:145;
for j being Nat st 1 <= j & j <= len (Replace (0* m),i,(x + y)) holds
(Replace (0* m),i,(x + y)) . j = ((Replace (0* m),i,x) + (Replace (0* m),i,y)) . j

proof

let j be Nat; :: thesis: ( 1 <= j & j <= len (Replace (0* m),i,(x + y)) implies (Replace (0* m),i,(x + y)) . j = ((Replace (0* m),i,x) + (Replace (0* m),i,y)) . j )
assume A4: ( 1 <= j & j <= len (Replace (0* m),i,(x + y)) ) ; :: thesis: (Replace (0* m),i,(x + y)) . j = ((Replace (0* m),i,x) + (Replace (0* m),i,y)) . j
reconsider j = j as Element of NAT by ORDINAL1:def 13;
A5: dom ((Replace (0* m),i,x) + (Replace (0* m),i,y)) = (dom (Replace (0* m),i,x)) /\ (dom (Replace (0* m),i,y)) by VALUED_1:def 1;
( j in dom (Replace (0* m),i,x) & j in dom (Replace (0* m),i,y) ) by A4, A2, FINSEQ_3:27;
then j in dom ((Replace (0* m),i,x) + (Replace (0* m),i,y)) by A5, XBOOLE_0:def 4;
then A6: ((Replace (0* m),i,x) + (Replace (0* m),i,y)) . j = ((Replace (0* m),i,x) . j) + ((Replace (0* m),i,y) . j) by VALUED_1:def 1;

per cases ( i = j or i <> j ) ;

suppose A7: i = j ; :: thesis: (Replace (0* m),i,(x + y)) . j = ((Replace (0* m),i,x) + (Replace (0* m),i,y)) . j

then ((Replace (0* m),i,x) + (Replace (0* m),i,y)) . j = x + ((Replace (0* m),i,y) . j) by A1, A6, Lm7
.= x + y by A1, A7, Lm7 ;
hence (Replace (0* m),i,(x + y)) . j = ((Replace (0* m),i,x) + (Replace (0* m),i,y)) . j by A1, A7, Lm7; :: thesis: verum

end;

suppose A8: i <> j ; :: thesis: (Replace (0* m),i,(x + y)) . j = ((Replace (0* m),i,x) + (Replace (0* m),i,y)) . j

then ((Replace (0* m),i,x) + (Replace (0* m),i,y)) . j = 0 + ((Replace (0* m),i,y) . j) by A4, A6, A2, Lm7
.= 0 by A4, A2, A8, Lm7 ;
hence (Replace (0* m),i,(x + y)) . j = ((Replace (0* m),i,x) + (Replace (0* m),i,y)) . j by A4, A2, A8, Lm7; :: thesis: verum

end;

end;

end;

hence Replace (0* m),i,(x + y) = (Replace (0* m),i,x) + (Replace (0* m),i,y) by A3, FINSEQ_1:18; :: thesis: verum