reserve i, j, m, n for Nat,
  z, B0 for set,
  f, x0 for real-valued FinSequence;

theorem Th28:
  1<=i & i<=n & i<>j implies |( Base_FinSeq(n,i),Base_FinSeq(n,j))| = 0
proof
  assume that
A1: 1<=i and
A2: i<=n and
A3: i<>j;
  set x1=Base_FinSeq(n,i),x2=Base_FinSeq(n,j);
A4: dom (0*n)=Seg len (n |-> (0 qua Real)) by FINSEQ_1:def 3
    .=Seg n by CARD_1:def 7;
A5: dom x2=Seg len x2 by FINSEQ_1:def 3
    .=Seg n by MATRIXR2:74;
A6: dom x1=Seg len x1 by FINSEQ_1:def 3
    .=Seg n by MATRIXR2:74;
A7: dom (<:x1,x2:>)=(dom x1) /\ dom x2 by FUNCT_3:def 7;
  dom (multreal)=[:REAL,REAL:] by FUNCT_2:def 1;
  then
A8: dom (multreal* (<:x1,x2:>)) = (<:x1,x2:>)"([:REAL,REAL:]) by RELAT_1:147
    .=Seg n by A7,A6,A5,RELSET_1:22;
  for x being object st x in dom (0*n)
holds (multreal* <:x1,x2:>).x= (0*n). x
  proof
    let x be object;
    assume
A9: x in dom (0*n);
    then reconsider nx=x as Element of NAT;
A10: (multreal* <:x1,x2:>).x=multreal.((<:x1,x2:>).x) by A4,A8,A9,FUNCT_1:12
      .=multreal.([x1.nx,x2.nx]) by A4,A7,A6,A5,A9,FUNCT_3:def 7;
A11: nx<=n by A4,A9,FINSEQ_1:1;
A12: 1<=nx by A4,A9,FINSEQ_1:1;
    per cases;
    suppose
A13:  nx=i;
      then
A14:  x1.nx=1 by A1,A2,MATRIXR2:75;
A15:  x2.nx=0 by A1,A2,A3,A13,MATRIXR2:76;
      multreal.([x1.nx,x2.nx])= multreal.(x1.nx,x2.nx)
        .=1*0 by A15,A14,BINOP_2:def 11
        .=0;
      hence (multreal* <:x1,x2:>).x= (0*n).x by A10;
    end;
    suppose
A16:  nx<>i;
      reconsider r=x2.nx as Element of REAL by XREAL_0:def 1;
A17:  x1.nx=0 by A12,A11,A16,MATRIXR2:76;
      multreal.([x1.nx,x2.nx])= multreal.(x1.nx,x2.nx)
        .=0 * r by A17,BINOP_2:def 11
        .=0;
      hence (multreal* <:x1,x2:>).x= (0*n).x by A10;
    end;
  end;
  then multreal* <:x1,x2:>= 0*n by A4,A8,FUNCT_1:2;
  then multreal .: (x1,x2)=0*n by FUNCOP_1:def 3;
  hence |( Base_FinSeq(n,i),Base_FinSeq(n,j) )|=0 by RVSUM_1:81;
end;
