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 Th17:
  x1<>x2 & x1<>x3 & x1<>x4 & x2<>x3 & x2<>x4 & x3<>x4 & 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)) & f1.x4 = 1 & (for z st z in A holds(z<>x4 implies f1.z = 0)) implies
  for a,b,c,d being Real
   st (RealFuncAdd(A)).((RealFuncAdd(A)).((RealFuncAdd(A)). ((
RealFuncExtMult(A)).[a,f],(RealFuncExtMult(A)).[b,g]), (RealFuncExtMult(A)).[c,
h]),(RealFuncExtMult(A)).[d,f1]) = RealFuncZero(A) holds a=0 & b=0 & c = 0 & d=
  0
proof
  set RM = RealFuncExtMult(A), RA = RealFuncAdd(A);
  assume that
A1: x1<>x2 and
A2: x1<>x3 and
A3: x1<>x4 and
A4: x2<>x3 and
A5: x2<>x4 and
A6: x3<>x4 and
A7: f.x1 = 1 and
A8: for z st z in A holds(z<>x1 implies f.z = 0) and
A9: g.x2 = 1 and
A10: for z st z in A holds(z<>x2 implies g.z = 0) and
A11: h.x3 = 1 and
A12: for z st z in A holds(z<>x3 implies h.z = 0) and
A13: f1.x4 = 1 and
A14: for z st z in A holds(z<>x4 implies f1.z = 0);
A15: f.x2=0 & h.x2=0 by A1,A4,A8,A12;
A16: g.x1=0 & h.x1=0 by A1,A2,A10,A12;
A17: f1.x1=0 by A3,A14;
A18: f1.x2=0 by A5,A14;
  let a,b,c,d be Real;
  assume
A19: RA.(RA.(RA.(RM.[a,f],RM.[b,g]),RM.[c,h]),RM.[d,f1]) = RealFuncZero( A);
   reconsider a,b,c,d as Element of REAL by XREAL_0:def 1;
A20: 0 = (RA.(RA.(RA.(RM.[a,f],RM.[b,g]),RM.[c,h]),RM.[d,f1])).x2
        by FUNCOP_1:7,A19
    .= (RA.(RA.(RM.[a,f],RM.[b,g]),RM.[c,h])).x2 + (RM.[d,f1]).x2 by FUNCSDOM:1
    .= (RA.(RM.[a,f],RM.[b,g])).x2 + (RM.[c,h]).x2 + (RM.[d,f1]).x2 by
FUNCSDOM:1
    .= (RM.[a,f]).x2 + (RM.[b,g]).x2 + (RM.[c,h]).x2 + (RM.[d,f1]).x2 by
FUNCSDOM:1
    .= (RM.[a,f]).x2 + (RM.[b,g]).x2 + (RM.[c,h]).x2 + d*(f1.x2) by FUNCSDOM:4
    .= (RM.[a,f]).x2 + (RM.[b,g]).x2 + c*(h.x2) + d*(f1.x2) by FUNCSDOM:4
    .= (RM.[a,f]).x2 + b*(g.x2) + c*(h.x2) + d*(f1.x2) by FUNCSDOM:4
    .= a*0 + b*1 + c*0 + d*0 by A9,A15,A18,FUNCSDOM:4
    .= b;
A21: f.x4=0 & g.x4=0 by A3,A5,A8,A10;
A22: h.x4=0 by A6,A12;
A23: f.x3=0 & g.x3=0 by A2,A4,A8,A10;
A24: f1.x3=0 by A6,A14;
A25: 0 = (RA.(RA.(RA.(RM.[a,f],RM.[b,g]),RM.[c,h]),RM.[d,f1])).x4 by A19,
FUNCOP_1:7
    .= (RA.(RA.(RM.[a,f],RM.[b,g]),RM.[c,h])).x4 + (RM.[d,f1]).x4 by FUNCSDOM:1
    .= (RA.(RM.[a,f], RM.[b,g])).x4 + (RM.[c,h]).x4 + (RM.[d,f1]).x4 by
FUNCSDOM:1
    .= (RM.[a,f]).x4 + (RM.[b,g]).x4 + (RM.[c,h]).x4 + (RM.[d,f1]).x4 by
FUNCSDOM:1
    .= (RM.[a,f]).x4 + (RM.[b,g]).x4 + (RM.[c,h]).x4 + d*(f1.x4) by FUNCSDOM:4
    .= (RM.[a,f]).x4 + (RM.[b,g]).x4 + c*(h.x4) + d*(f1.x4) by FUNCSDOM:4
    .= (RM.[a,f]).x4 + b*(g.x4) + c*(h.x4) + d*(f1.x4) by FUNCSDOM:4
    .= a*0 + b*0 + c*0 + d*1 by A13,A21,A22,FUNCSDOM:4
    .= d;
A26: 0 = (RA.(RA.(RA.(RM.[a,f],RM.[b,g]),RM.[c,h]),RM.[d,f1])).x3 by A19,
FUNCOP_1:7
    .= (RA.(RA.(RM.[a,f], RM.[b,g]),RM.[c,h])).x3 + (RM.[d,f1]).x3 by
FUNCSDOM:1
    .= (RA.(RM.[a,f],RM.[b,g])).x3 + (RM.[c,h]).x3 + (RM.[d,f1]).x3 by
FUNCSDOM:1
    .= (RM.[a,f]).x3 + (RM.[b,g]).x3 + (RM.[c,h]).x3 + (RM.[d,f1]).x3 by
FUNCSDOM:1
    .= (RM.[a,f]).x3 + (RM.[b,g]).x3 + (RM.[c,h]).x3 + d*(f1.x3) by FUNCSDOM:4
    .= (RM.[a,f]).x3 + (RM.[b,g]).x3 + c*(h.x3) + d*(f1.x3) by FUNCSDOM:4
    .= (RM.[a,f]).x3 + b*(g.x3) + c*(h.x3) + d*(f1.x3) by FUNCSDOM:4
    .= a*0 + b*0 + c*1 + d*0 by A11,A23,A24,FUNCSDOM:4
    .= c;
  0 = (RA.(RA.(RA.(RM.[a,f],RM.[b,g]),RM.[c,h]),RM.[d,f1])).x1 by A19,
FUNCOP_1:7
    .= (RA.(RA.(RM.[a,f],RM.[b,g]),RM.[c,h])).x1 + (RM.[d,f1]).x1 by FUNCSDOM:1
    .= (RA.(RM.[a,f],RM.[b,g])).x1 + (RM.[c,h]).x1 + (RM.[d,f1]).x1 by
FUNCSDOM:1
    .= (RM.[a,f]).x1 + (RM.[b,g]).x1 + (RM.[c,h]).x1 + (RM.[d,f1]).x1 by
FUNCSDOM:1
    .= (RM.[a,f]).x1 + (RM.[b,g]).x1 + (RM.[c,h]).x1 + d*(f1.x1) by FUNCSDOM:4
    .= (RM.[a,f]).x1 + (RM.[b,g]).x1 + c*(h.x1) + d*(f1.x1) by FUNCSDOM:4
    .= (RM.[a,f]).x1 + b*(g.x1) + c*(h.x1) + d*(f1.x1) by FUNCSDOM:4
    .= a*1 + b*0 + c*0 + d*0 by A7,A16,A17,FUNCSDOM:4
    .= a;
  hence thesis by A20,A26,A25;
end;
