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 Th11:
  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 a,b,c being Real
  st (RealFuncAdd(A)).((RealFuncAdd(A)). ((RealFuncExtMult(A)).[a,f],(
RealFuncExtMult(A)).[b,g]), (RealFuncExtMult(A)).[c,h]) = RealFuncZero(A)
  holds
  a=0 & b=0 & c = 0
proof
  set RM = RealFuncExtMult(A), RA = RealFuncAdd(A);
  assume that
A1: x1<>x2 and
A2: x1<>x3 and
A3: x2<>x3 and
A4: f.x1 = 1 and
A5: for z st z in A holds(z<>x1 implies f.z = 0) and
A6: g.x2 = 1 and
A7: for z st z in A holds(z<>x2 implies g.z = 0) and
A8: h.x3 = 1 and
A9: for z st z in A holds(z<>x3 implies h.z = 0);
A10: f.x2=0 & h.x2=0 by A1,A3,A5,A9;
  let a,b,c be Real;
  assume
A11: RA.(RA.(RM.[a,f],RM.[b,g]),RM.[c,h]) = RealFuncZero(A);
   reconsider a,b,c as Element of REAL by XREAL_0:def 1;
A12: 0 = (RA.(RA.(RM.[a,f],RM.[b,g]), RM.[c,h])).x2 by FUNCOP_1:7,A11
    .= (RA.(RM.[a,f],RM.[b,g])).x2 + (RM.[c,h]).x2 by FUNCSDOM:1
    .= ((RM.[a,f]).x2) + ((RM.[b,g]).x2) + (RM.[c,h]).x2 by FUNCSDOM:1
    .= ((RM.[a,f]).x2) + ((RM.[b,g]).x2) + c*(h.x2) by FUNCSDOM:4
    .= ((RM.[a,f]).x2) + b*(g.x2) + c*(h.x2) by FUNCSDOM:4
    .= a*0 + b*1 + c*0 by A6,A10,FUNCSDOM:4
    .= b;
A13: g.x1=0 & h.x1=0 by A1,A2,A7,A9;
A14: f.x3=0 & g.x3=0 by A2,A3,A5,A7;
A15: 0 = (RA.(RA.(RM.[a,f],RM.[b,g]), RM.[c,h])).x3 by A11,FUNCOP_1:7
    .= (RA.(RM.[a,f],RM.[b,g])).x3 + (RM.[c,h]).x3 by FUNCSDOM:1
    .= ((RM.[a,f]).x3) + ((RM.[b,g]).x3) + (RM.[c,h]).x3 by FUNCSDOM:1
    .= ((RM.[a,f]).x3) + ((RM.[b,g]).x3) + c*(h.x3) by FUNCSDOM:4
    .= ((RM.[a,f]).x3) + b*(g.x3) + c*(h.x3) by FUNCSDOM:4
    .= a*0 + b*0 + c*1 by A8,A14,FUNCSDOM:4
    .= c;
  0 = (RA.(RA.(RM.[a,f],RM.[b,g]),RM.[c,h])).x1 by A11,FUNCOP_1:7
    .= (RA.(RM.[a,f],RM.[b,g])).x1 + (RM.[c,h]).x1 by FUNCSDOM:1
    .= ((RM.[a,f]).x1) + ((RM.[b,g]).x1) + (RM.[c,h]).x1 by FUNCSDOM:1
    .= ((RM.[a,f]).x1) + ((RM.[b,g]).x1) + c*(h.x1) by FUNCSDOM:4
    .= ((RM.[a,f]).x1) + b*(g.x1) + c*(h.x1) by FUNCSDOM:4
    .= a*1 + b*0 + c*0 by A4,A13,FUNCSDOM:4
    .= a;
  hence thesis by A12,A15;
end;
