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 Th19:
  A = {x1,x2,x3,x4} & 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 h9 being Element of Funcs(A,REAL)
  ex a,b,c ,d being Real
   st h9 = (RealFuncAdd(A)).((RealFuncAdd(A)).((RealFuncAdd(A)). ((
RealFuncExtMult(A)).[a,f],(RealFuncExtMult(A)).[b,g]), (RealFuncExtMult(A)).[c,
  h]),(RealFuncExtMult(A)).[d,f1])
proof
  assume that
A1: A = {x1,x2,x3,x4} and
A2: x1<>x2 and
A3: x1<>x3 and
A4: x1<>x4 and
A5: x2<>x3 and
A6: x2<>x4 and
A7: x3<>x4 and
A8: f.x1 = 1 and
A9: for z st z in A holds(z<>x1 implies f.z = 0) and
A10: g.x2 = 1 and
A11: for z st z in A holds(z<>x2 implies g.z = 0) and
A12: h.x3 = 1 and
A13: for z st z in A holds(z<>x3 implies h.z = 0) and
A14: f1.x4 = 1 and
A15: for z st z in A holds(z<>x4 implies f1.z = 0);
A16: f.x4=0 & g.x4=0 by A4,A6,A9,A11;
A17: f1.x3=0 by A7,A15;
A18: f1.x2=0 by A6,A15;
A19: f.x2=0 & h.x2=0 by A2,A5,A9,A13;
A20: f1.x1=0 by A4,A15;
A21: g.x1=0 & h.x1=0 by A2,A3,A11,A13;
A22: h.x4=0 by A7,A13;
  let h9 be Element of Funcs(A,REAL);
  take a = h9.x1, b = h9.x2, c = h9.x3, d = h9.x4;
A23: f.x3=0 & g.x3=0 by A3,A5,A9,A11;
  now
    let x be Element of A;
A24: x = x1 or x = x2 or x = x3 or x = x4 by A1,ENUMSET1:def 2;
A25: ((RealFuncAdd(A)).((RealFuncAdd(A)).((RealFuncAdd(A)). ((
RealFuncExtMult(A)).[a,f],(RealFuncExtMult(A)).[b,g]), (RealFuncExtMult(A)).[c,
h]),(RealFuncExtMult(A)).[d,f1])).x2 = ((RealFuncAdd(A)).((RealFuncAdd(A)).((
RealFuncExtMult(A)).[a,f], (RealFuncExtMult(A)).[b,g]),(RealFuncExtMult(A)).[c,
    h])).x2 + ((RealFuncExtMult(A)).[d,f1]).x2 by FUNCSDOM:1
      .= ((RealFuncAdd(A)).((RealFuncExtMult(A)).[a,f], (RealFuncExtMult(A))
.[b,g])).x2 + ((RealFuncExtMult(A)).[c,h]).x2 + ((RealFuncExtMult(A)).[d,f1]).
    x2 by FUNCSDOM:1
      .= ((RealFuncExtMult(A)).[a,f]).x2 + ((RealFuncExtMult(A)).[b,g]).x2 +
((RealFuncExtMult(A)).[c,h]).x2 + ((RealFuncExtMult(A)).[d,f1]).x2 by
FUNCSDOM:1
      .= ((RealFuncExtMult(A)).[a,f]).x2 + ((RealFuncExtMult(A)).[b,g]).x2 +
    ((RealFuncExtMult(A)).[c,h]).x2 + d*(f1.x2) by FUNCSDOM:4
      .= ((RealFuncExtMult(A)).[a,f]).x2 + ((RealFuncExtMult(A)).[b,g]).x2 +
    c*(h.x2) + d*(f1.x2) by FUNCSDOM:4
      .= ((RealFuncExtMult(A)).[a,f]).x2 + b*(g.x2) + c*(h.x2) + d*(f1.x2)
    by FUNCSDOM:4
      .= a*0 + b*1 + c*0 + d*0 by A10,A19,A18,FUNCSDOM:4
      .= h9.x2;
A26: ((RealFuncAdd(A)).((RealFuncAdd(A)).((RealFuncAdd(A)). ((
RealFuncExtMult(A)).[a,f],(RealFuncExtMult(A)).[b,g]), (RealFuncExtMult(A)).[c,
h]),(RealFuncExtMult(A)).[d,f1])).x4 = ((RealFuncAdd(A)).((RealFuncAdd(A)).((
RealFuncExtMult(A)).[a,f], (RealFuncExtMult(A)).[b,g]),(RealFuncExtMult(A)).[c,
    h])).x4 + ((RealFuncExtMult(A)).[d,f1]).x4 by FUNCSDOM:1
      .= ((RealFuncAdd(A)).((RealFuncExtMult(A)).[a,f], (RealFuncExtMult(A))
.[b,g])).x4 + ((RealFuncExtMult(A)).[c,h]).x4 + ((RealFuncExtMult(A)).[d,f1]).
    x4 by FUNCSDOM:1
      .= ((RealFuncExtMult(A)).[a,f]).x4 + ((RealFuncExtMult(A)).[b,g]).x4 +
((RealFuncExtMult(A)).[c,h]).x4 + ((RealFuncExtMult(A)).[d,f1]).x4 by
FUNCSDOM:1
      .= ((RealFuncExtMult(A)).[a,f]).x4 + ((RealFuncExtMult(A)).[b,g]).x4 +
    ((RealFuncExtMult(A)).[c,h]).x4 + d*(f1.x4) by FUNCSDOM:4
      .= ((RealFuncExtMult(A)).[a,f]).x4 + ((RealFuncExtMult(A)).[b,g]).x4 +
    c*(h.x4) + d*(f1.x4) by FUNCSDOM:4
      .= ((RealFuncExtMult(A)).[a,f]).x4 + b*(g.x4) + c*(h.x4) + d*(f1.x4)
    by FUNCSDOM:4
      .= a*0 + b*0 + c*0 + d*1 by A14,A16,A22,FUNCSDOM:4
      .= h9.x4;
A27: ((RealFuncAdd(A)).((RealFuncAdd(A)).((RealFuncAdd(A)). ((
RealFuncExtMult(A)).[a,f],(RealFuncExtMult(A)).[b,g]), (RealFuncExtMult(A)).[c,
h]),(RealFuncExtMult(A)).[d,f1])).x3 = ((RealFuncAdd(A)).((RealFuncAdd(A)).((
RealFuncExtMult(A)).[a,f], (RealFuncExtMult(A)).[b,g]),(RealFuncExtMult(A)).[c,
    h])).x3 + ((RealFuncExtMult(A)).[d,f1]).x3 by FUNCSDOM:1
      .= ((RealFuncAdd(A)).((RealFuncExtMult(A)).[a,f], (RealFuncExtMult(A))
.[b,g])).x3 + ((RealFuncExtMult(A)).[c,h]).x3 + ((RealFuncExtMult(A)).[d,f1]).
    x3 by FUNCSDOM:1
      .= ((RealFuncExtMult(A)).[a,f]).x3 + ((RealFuncExtMult(A)).[b,g]).x3 +
((RealFuncExtMult(A)).[c,h]).x3 + ((RealFuncExtMult(A)).[d,f1]).x3 by
FUNCSDOM:1
      .= ((RealFuncExtMult(A)).[a,f]).x3 + ((RealFuncExtMult(A)).[b,g]).x3 +
    ((RealFuncExtMult(A)).[c,h]).x3 + d*(f1.x3) by FUNCSDOM:4
      .= ((RealFuncExtMult(A)).[a,f]).x3 + ((RealFuncExtMult(A)).[b,g]).x3 +
    c*(h.x3) + d*(f1.x3) by FUNCSDOM:4
      .= ((RealFuncExtMult(A)).[a,f]).x3 + b*(g.x3) + c*(h.x3) + d*(f1.x3)
    by FUNCSDOM:4
      .= a*0 + b*0 + c*1 + d*0 by A12,A23,A17,FUNCSDOM:4
      .= h9.x3;
    ((RealFuncAdd(A)).((RealFuncAdd(A)).((RealFuncAdd(A)). ((
RealFuncExtMult(A)).[a,f],(RealFuncExtMult(A)).[b,g]), (RealFuncExtMult(A)).[c,
h]),(RealFuncExtMult(A)).[d,f1])).x1 = ((RealFuncAdd(A)).((RealFuncAdd(A)).((
RealFuncExtMult(A)).[a,f], (RealFuncExtMult(A)).[b,g]),(RealFuncExtMult(A)).[c,
    h])).x1 + ((RealFuncExtMult(A)).[d,f1]).x1 by FUNCSDOM:1
      .= ((RealFuncAdd(A)).((RealFuncExtMult(A)).[a,f], (RealFuncExtMult(A))
.[b,g])).x1 + ((RealFuncExtMult(A)).[c,h]).x1 + ((RealFuncExtMult(A)).[d,f1]).
    x1 by FUNCSDOM:1
      .= ((RealFuncExtMult(A)).[a,f]).x1 + ((RealFuncExtMult(A)).[b,g]).x1 +
((RealFuncExtMult(A)).[c,h]).x1 + ((RealFuncExtMult(A)).[d,f1]).x1 by
FUNCSDOM:1
      .= ((RealFuncExtMult(A)).[a,f]).x1 + ((RealFuncExtMult(A)).[b,g]).x1 +
    ((RealFuncExtMult(A)).[c,h]).x1 + d*(f1.x1) by FUNCSDOM:4
      .= ((RealFuncExtMult(A)).[a,f]).x1 + ((RealFuncExtMult(A)).[b,g]).x1 +
    c*(h.x1) + d*(f1.x1) by FUNCSDOM:4
      .= ((RealFuncExtMult(A)).[a,f]).x1 + b*(g.x1) + c*(h.x1) + d*(f1.x1)
    by FUNCSDOM:4
      .= a*1 + b*0 + c*0 + d*0 by A8,A21,A20,FUNCSDOM:4
      .= h9.x1;
    hence h9.x = ((RealFuncAdd(A)).((RealFuncAdd(A)).((RealFuncAdd(A)). ((
RealFuncExtMult(A)).[a,f],(RealFuncExtMult(A)).[b,g]), (RealFuncExtMult(A)).[c,
    h]),(RealFuncExtMult(A)).[d,f1])).x by A24,A25,A27,A26;
  end;
  hence thesis by FUNCT_2:63;
end;
