reserve X for non empty set;
reserve Y for RealLinearSpace;
reserve f,g,h for Element of Funcs(X,the carrier of Y);
reserve a,b for Real;
reserve u,v,w for VECTOR of RLSStruct(#Funcs(X,the carrier of Y), (FuncZero(X,
    Y)),FuncAdd(X,Y),FuncExtMult(X,Y)#);

theorem Th10:
  RLSStruct(#Funcs(X,the carrier of Y), (FuncZero(X,Y)),FuncAdd(X,
    Y),FuncExtMult(X,Y)#) is RealLinearSpace
proof
A1: for a,b being Real,v holds (a*b)*v = a*(b*v)
  by Th8;
A2: for a,b being Real,v holds (a+b)*v = a*v + b*v
  by Th9;
  set IT = RLSStruct(#Funcs(X,the carrier of Y), (FuncZero(X,Y)),FuncAdd(X,Y),
    FuncExtMult(X,Y)#);
A3: (u+v)+w = u+(v+w) by Th4;
A4: IT is right_complementable
  proof
    let u;
    reconsider u9 = u as Element of Funcs(X,the carrier of Y);
    reconsider w = (FuncExtMult(X,Y)).[-jj,u9] as VECTOR of IT;
    take w;
    thus thesis by Th6;
  end;
A5: for a being Real,u,v holds a*(u+v) = a*u + a *v
  by Lm2;
A6: 1*v = v by Th7;
A7: u+0.IT = u
  proof
    reconsider u9=u as Element of Funcs(X,the carrier of Y);
    thus u+0.IT = (FuncAdd(X,Y)).(FuncZero(X,Y),u9) by Th3
      .= u by Th5;
  end;
  u+v = v+u by Th3;
  hence thesis by A3,A7,A4,A5,A2,A1,A6,RLVECT_1:def 2,def 3,def 4,def 5,def 6
,def 7,def 8;
end;
