reserve V for RealLinearSpace,
  o,p,q,r,s,u,v,w,y,y1,u1,v1,w1,u2,v2,w2 for Element of V,
  a,b,c,d,a1,b1,c1,d1,a2,b2,c2,d2,a3,b3,c3,d3 for Real,
  z for set;
reserve A for non empty set;
reserve f,g,h,f1 for Element of Funcs(A,REAL);
reserve x1,x2,x3,x4 for Element of A;

theorem Th13:
  A = {x1,x2,x3} & x1<>x2 & x1<>x3 & x2<>x3 & f.x1 = 1 & (for z st
z in A holds (z<>x1 implies f.z = 0)) & g.x2 = 1 & (for z st z in A holds (z<>
x2 implies g.z = 0)) & h.x3 = 1 & (for z st z in A holds (z<>x3 implies h.z = 0
  )) implies for h9 being Element of Funcs(A,REAL) holds
  ex a,b,c being Real st h9 = (
RealFuncAdd(A)).((RealFuncAdd(A)). ((RealFuncExtMult(A)).[a,f],(RealFuncExtMult
  (A)).[b,g]), (RealFuncExtMult(A)).[c,h])
proof
  assume that
A1: A = {x1,x2,x3} and
A2: x1<>x2 and
A3: x1<>x3 and
A4: x2<>x3 and
A5: f.x1 = 1 and
A6: for z st z in A holds(z<>x1 implies f.z = 0) and
A7: g.x2 = 1 and
A8: for z st z in A holds(z<>x2 implies g.z = 0) and
A9: h.x3 = 1 and
A10: for z st z in A holds(z<>x3 implies h.z = 0);
A11: g.x1=0 & h.x1=0 by A2,A3,A8,A10;
A12: f.x2=0 & h.x2=0 by A2,A4,A6,A10;
  let h9 be Element of Funcs(A,REAL);
  take a = h9.x1, b = h9.x2, c = h9.x3;
A13: f.x3=0 & g.x3=0 by A3,A4,A6,A8;
  now
    let x be Element of A;
A14: x = x1 or x = x2 or x = x3 by A1,ENUMSET1:def 1;
A15: ((RealFuncAdd(A)).((RealFuncAdd(A)). ((RealFuncExtMult(A)).[a,f],(
RealFuncExtMult(A)).[b,g]), (RealFuncExtMult(A)).[c,h])).x2 = ((RealFuncAdd(A))
    . ((RealFuncExtMult(A)).[a,f],(RealFuncExtMult(A)).[b,g])).x2 + ((
    RealFuncExtMult(A)).[c,h]).x2 by FUNCSDOM:1
      .= (((RealFuncExtMult(A)).[a,f]).x2) + (((RealFuncExtMult(A)).[b,g]).
    x2) + ((RealFuncExtMult(A)).[c,h]).x2 by FUNCSDOM:1
      .= (((RealFuncExtMult(A)).[a,f]).x2) + (((RealFuncExtMult(A)).[b,g]).
    x2) + c*(h.x2) by FUNCSDOM:4
      .= (((RealFuncExtMult(A)).[a,f]).x2) + b*(g.x2) + c*(h.x2) by FUNCSDOM:4
      .= a*0 + b*1 + c*0 by A7,A12,FUNCSDOM:4
      .= h9.x2;
A16: ((RealFuncAdd(A)).((RealFuncAdd(A)). ((RealFuncExtMult(A)).[a,f],(
RealFuncExtMult(A)).[b,g]), (RealFuncExtMult(A)).[c,h])).x3 = ((RealFuncAdd(A))
    . ((RealFuncExtMult(A)).[a,f],(RealFuncExtMult(A)).[b,g])).x3 + ((
    RealFuncExtMult(A)).[c,h]).x3 by FUNCSDOM:1
      .= (((RealFuncExtMult(A)).[a,f]).x3) + (((RealFuncExtMult(A)).[b,g]).
    x3) + ((RealFuncExtMult(A)).[c,h]).x3 by FUNCSDOM:1
      .= (((RealFuncExtMult(A)).[a,f]).x3) + (((RealFuncExtMult(A)).[b,g]).
    x3) + c*(h.x3) by FUNCSDOM:4
      .= (((RealFuncExtMult(A)).[a,f]).x3) + b*(g.x3) + c*(h.x3) by FUNCSDOM:4
      .= a*0 + b*0 + c*1 by A9,A13,FUNCSDOM:4
      .= h9.x3;
    ((RealFuncAdd(A)).((RealFuncAdd(A)). ((RealFuncExtMult(A)).[a,f],(
RealFuncExtMult(A)).[b,g]), (RealFuncExtMult(A)).[c,h])).x1 = ((RealFuncAdd(A))
    . ((RealFuncExtMult(A)).[a,f],(RealFuncExtMult(A)).[b,g])).x1 + ((
    RealFuncExtMult(A)).[c,h]).x1 by FUNCSDOM:1
      .= (((RealFuncExtMult(A)).[a,f]).x1) + (((RealFuncExtMult(A)).[b,g]).
    x1) + ((RealFuncExtMult(A)).[c,h]).x1 by FUNCSDOM:1
      .= (((RealFuncExtMult(A)).[a,f]).x1) + (((RealFuncExtMult(A)).[b,g]).
    x1) + c*(h.x1) by FUNCSDOM:4
      .= (((RealFuncExtMult(A)).[a,f]).x1) + b*(g.x1) + c*(h.x1) by FUNCSDOM:4
      .= a*1 + b*0 + c*0 by A5,A11,FUNCSDOM:4
      .= h9.x1;
    hence
    h9.x = ((RealFuncAdd(A)).((RealFuncAdd(A)). ((RealFuncExtMult(A)).[a,
    f],(RealFuncExtMult(A)).[b,g]), (RealFuncExtMult(A)).[c,h])).x by A14,A15
,A16;
  end;
  hence thesis by FUNCT_2:63;
end;
