reserve f for PartFunc of REAL-NS 1,REAL-NS 1;
reserve g for PartFunc of REAL,REAL;
reserve x for Point of REAL-NS 1;
reserve y for Real;
reserve m,n for non zero Nat;
reserve i,j for Nat;
reserve f for PartFunc of REAL-NS n,REAL-NS 1;
reserve g for PartFunc of REAL n,REAL;
reserve x for Point of REAL-NS n;
reserve y for Element of REAL n;
reserve X for set;
reserve r for Real;
reserve f,f1,f2 for PartFunc of REAL-NS m,REAL-NS n;
reserve g,g1,g2 for PartFunc of REAL n,REAL;
reserve h for PartFunc of REAL m,REAL n;
reserve x for Point of REAL-NS m;
reserve y for Element of REAL n;
reserve z for Element of REAL m;

theorem Th27:
  r(#)(f*reproj(i,x)) = (r(#)f)*reproj(i,x)
proof
A1: dom(r(#)f) = dom f by VFUNCT_1:def 4;
A2: dom(r(#)(f*reproj(i,x))) =dom(f*reproj(i,x)) by VFUNCT_1:def 4;
A3: dom(reproj(i,x))=the carrier of REAL-NS 1 by FUNCT_2:def 1;
  for s be Element of REAL-NS 1 holds s in dom((r(#)f)*reproj(i,x)) iff s
  in dom(f*reproj(i,x))
  proof
    let s be Element of REAL-NS 1;
    s in dom((r(#)f)*reproj(i,x)) iff reproj(i,x).s in dom(r(#)f) by A3,
FUNCT_1:11;
    hence thesis by A1,A3,FUNCT_1:11;
  end;
  then for s be object
    holds s in dom(r(#)(f*reproj(i,x))) iff s in dom((r(#)f)*
  reproj(i,x)) by A2;
  then
A4: dom(r(#)(f*reproj(i,x))) =dom((r(#)f)*reproj(i,x)) by TARSKI:2;
A5: for s be Element of REAL-NS 1 holds s in dom((r(#)f)*reproj(i,x)) iff
  reproj(i,x).s in dom(r(#)f)
  proof
    let s be Element of REAL-NS 1;
    dom(reproj(i,x))=the carrier of REAL-NS 1 by FUNCT_2:def 1;
    hence thesis by FUNCT_1:11;
  end;
  for z being Element of REAL-NS 1 st z in dom(r(#)(f*reproj(i,x))) holds
  (r(#)(f*reproj(i,x))).z = ((r(#)f)*reproj(i,x)).z
  proof
    let z be Element of REAL-NS 1;
    assume
A6: z in dom(r(#)(f*reproj(i,x)));
    then
A7: z in dom(f*reproj(i,x)) by VFUNCT_1:def 4;
A8: reproj(i,x).z in dom f by A1,A5,A4,A6;
    then
A9: f/.(reproj(i,x).z) = f.(reproj(i,x).z) by PARTFUN1:def 6
      .= (f*reproj(i,x)).z by A7,FUNCT_1:12
      .= (f*reproj(i,x))/.z by A7,PARTFUN1:def 6;
A10: (r(#)(f*reproj(i,x))).z =(r(#)(f*reproj(i,x)))/.z by A6,PARTFUN1:def 6
      .= r * f/.(reproj(i,x).z) by A6,A9,VFUNCT_1:def 4;
    ((r(#)f)*reproj(i,x)).z = (r(#)f).(reproj(i,x).z) by A4,A6,FUNCT_1:12
      .= (r(#)f)/.(reproj(i,x).z) by A1,A8,PARTFUN1:def 6
      .= r * f/.(reproj(i,x).z) by A1,A8,VFUNCT_1:def 4;
    hence thesis by A10;
  end;
  hence thesis by A4,PARTFUN1:5;
end;
