reserve n for Nat,
        p,p1,p2 for Point of TOP-REAL n,
        x for Real;
reserve n,m for non zero Nat;
reserve i,j for Nat;
reserve f for PartFunc of REAL-NS m,REAL-NS n;
reserve g for PartFunc of REAL m,REAL n;
reserve h for PartFunc of REAL m,REAL;
reserve x for Point of REAL-NS m;
reserve y for Element of REAL m;
reserve X for set;

theorem Th10:
for x,a be Real,i be Nat st
 1 <= i & i <= m holds 0*m+*(i,a*x) = a*(Replace(0*m,i,x))
proof
   let x,a be Real;
   let i be Nat;
   assume A1: 1 <= i & i <= m;
    reconsider ax = a*x as Real;
A2:len Replace(0*m,i,ax) = m & len Replace(0*m,i,x) = m by Lm6;
then A3:len(a*Replace(0*m,i,x)) = len(Replace(0*m,i,ax)) by RVSUM_1:117;
    for j be Nat st 1 <= j & j <= len(Replace(0*m,i,ax)) holds
    Replace(0*m,i,ax).j = (a*Replace(0*m,i,x)).j
   proof
    let j be Nat;
    assume A4: 1 <= j & j <= len(Replace(0*m,i,ax));
    reconsider j as Nat;
    per cases;
    suppose A5: i = j;
     then Replace(0*m,i,ax).j = a*x by A1,Lm7
       .= a*(Replace(0*m,i,x).j) by A1,A5,Lm7;
     hence thesis by RVSUM_1:44;
    end;
    suppose A6: i <> j;
     then Replace(0*m,i,x).j = 0 by A2,A4,Lm7;
     then Replace(0*m,i,ax).j = a*(Replace(0*m,i,x).j) by A2,A4,A6,Lm7;
     hence thesis by RVSUM_1:44;
    end;
   end;
   hence thesis by A3;
end;
