
theorem Th26:
  for A being Subset of TOP-REAL 3
  for B being convex non empty Subset of TOP-REAL 2
  for r being Real
  for x being Element of TOP-REAL 3 st
  A = {x where x is Element of TOP-REAL 3:|[x`1,x`2]| in B & x`3 = r}
  holds A is non empty convex
  proof
    let A be Subset of TOP-REAL 3;
    let B be convex non empty Subset of TOP-REAL 2;
    let r be Real;
    let x be Element of TOP-REAL 3;
    assume
A1: A = {x where x is Element of TOP-REAL 3:|[x`1,x`2]| in B & x`3 = r};
A2: for z be Element of TOP-REAL 3 holds z in A iff |[z`1,z`2]| in B & z`3 = r
    proof
      let z be Element of TOP-REAL 3;
      hereby
        assume z in A;
        then ex z9 be Element of TOP-REAL 3 st z = z9 & |[z9`1,z9`2]| in B &
          z9`3 = r by A1;
        hence |[z`1,z`2]| in B & z`3 = r;
      end;
      assume |[z`1,z`2]| in B & z`3 = r;
      hence z in A by A1;
    end;
    set y = the Element of B;
    reconsider z = |[y`1,y`2,r]| as Element of TOP-REAL 3;
    z = |[z`1,z`2,z`3]| by EUCLID_5:3;
    then
A3: z`1 = y`1 & z`2 = y`2 & z`3 = r by FINSEQ_1:78;
    y in B;
    then |[z`1,z`2]| in B & z`3 = r by A3,EUCLID:53;
    then z in A by A1;
    hence A is non empty;
    now
      let u,v be Element of TOP-REAL 3;
      let s be Real;
      assume that
A4:   0 < s < 1 and
A5:   u in A and
A6:   v in A;
      reconsider w = s * u + (1 - s) * v as Element of TOP-REAL 3;
      now
        reconsider su = s * u,sv = (1 - s) * v as Element of TOP-REAL 3;
        su = |[s * u`1,s * u`2,s * u`3]| &
          sv = |[(1-s)*v`1,(1-s)*v`2,(1-s)*v`3]| by EUCLID_5:7;
        then
A7:     |[s * u`1 + (1-s)*v`1,s*u`2+(1-s)*v`2,s*u`3+(1-s)*v`3]|
          = w by EUCLID_5:6
         .= |[w`1,w`2,w`3]| by EUCLID_5:3;
        then
A8:     w`1 = s * u`1 + (1-s)*v`1 & w`2=s*u`2+(1-s)*v`2 & w`3=s*u`3+(1-s)*v`3
          by FINSEQ_1:78;
        u`3 = r & v`3 = r by A5,A6,A2;
        hence w`3 = r by A7,FINSEQ_1:78;
        reconsider u9 = |[u`1,u`2]|,v9 = |[v`1,v`2]|,w9 = |[w`1,w`2]|
          as Element of TOP-REAL 2;
        now
          thus u9 in B & v9 in B by A2,A6,A5;
          thus |[w`1,w`2]| = |[s*u`1,s*u`2]| + |[(1-s)*v`1,(1-s)*v`2]|
                             by A8,EUCLID:56
                          .= s * |[u`1,u`2]| + |[(1-s)*v`1,(1-s)*v`2]|
                             by EUCLID:58
                          .= s * u9 + (1 - s) * v9 by EUCLID:58;
        end;
        hence |[w`1,w`2]| in B by A4,CONVEX1:def 2;
      end;
      hence s * u + (1 - s) * v in A by A1;
    end;
    hence A is convex by CONVEX1:def 2;
  end;
