
theorem Th6:
for A be Subset of [:RNS_Real,RNS_Real,RNS_Real:] st
   (for a,b,c be Real st [a,b,c] in A holds
     ex Rx be real-membered set st Rx is non empty bounded_above &
        Rx = {r where r is Real : 0 < r &
            [: ].a-r,a+r.[, ].b-r,b+r.[, ].c-r,c+r.[ :] c= A})
holds
  ex F be Function of A,REAL st
    for a,b,c be Real st [a,b,c] in A holds
     ex Rx be real-membered set st Rx is non empty bounded_above
       & Rx = {r where r is Real :0 < r
             & [: ].a-r,a+r.[, ].b-r,b+r.[, ].c-r,c+r.[ :] c= A}
       & F.([a,b,c]) = (upper_bound Rx) / 2
proof
    let A be Subset of [:RNS_Real,RNS_Real,RNS_Real:];
    assume
A1:  for a,b,c be Real st [a,b,c] in A holds
      ex Rx be real-membered set st Rx is non empty bounded_above
       & Rx = {r where r is Real : 0 < r
            & [: ].a-r,a+r.[, ].b-r,b+r.[, ].c-r,c+r.[ :] c= A};

    defpred P[object,object] means
     ex a,b,c be Real, Rx be real-membered set st $1 = [a,b,c]
       & Rx is non empty bounded_above
       & Rx = {r where r is Real :0 < r
            & [: ].a-r,a+r.[, ].b-r,b+r.[, ].c-r,c+r.[ :] c= A}
       & $2= (upper_bound Rx) / 2;

A2: for x being object st x in A ex y being object st y in REAL & P[x,y]
    proof
     let x be object;
     assume A3: x in A; then
     consider a1 be Point of RNS_Real, b1 be Point of RNS_Real,
       c1 be Point of RNS_Real such that
A4:   x = [a1,b1,c1] by PRVECT_4:9;

     reconsider a = a1, b = b1, c = c1 as Real;
     consider Rx be real-membered set such that
A5:   Rx is non empty bounded_above
    & Rx = {r where r is Real :0< r &
             [: ].a-r,a+r.[, ].b-r,b+r.[, ].c-r,c+r.[ :] c= A} by A1,A3,A4;

     take y = (upper_bound Rx)/2;
     thus y in REAL by XREAL_0:def 1;
     thus thesis by A4,A5;
    end;

    consider F being Function of A,REAL such that
A6:  for x being object st x in A holds P[x,F.x] from FUNCT_2:sch 1(A2);
    take F;
    thus for a,b,c be Real st [a,b,c] in A holds
      ex Rx be real-membered set st Rx is non empty bounded_above
        & Rx = {r where r is Real :0 < r
             & [: ].a-r,a+r.[, ].b-r,b+r.[, ].c-r,c+r.[ :] c= A}
        & F.([a,b,c]) =(upper_bound Rx)/2
    proof
     let a1,b1,c1 be Real;
     assume [a1,b1,c1] in A; then
     consider a,b,c be Real, Rx be real-membered set such that
A7:   [a1,b1,c1] = [a,b,c]
    & Rx is non empty bounded_above
    & Rx = {r where r is Real : 0 < r
         & [: ].a-r,a+r.[, ].b-r,b+r.[, ].c-r,c+r.[ :] c= A}
    & F.([a1,b1,c1]) = (upper_bound Rx)/2 by A6;
     a1=a & b1=b & c1=c by XTUPLE_0:3,A7;
     hence thesis by A7;
    end;
end;
