 reserve j for set;
 reserve p,r for Real;
 reserve S,T,F for RealNormSpace;
 reserve x0 for Point of S;
 reserve g for PartFunc of S,T;
 reserve c for constant sequence of S;
 reserve R for RestFunc of S,T;
 reserve G for RealNormSpace-Sequence;
 reserve i for Element of dom G;
 reserve f for PartFunc of product G,F;
 reserve x for Element of product G;
reserve G for RealNormSpace-Sequence;
reserve F for RealNormSpace;
reserve i for Element of dom G;
reserve f,f1,f2 for PartFunc of product G, F;
reserve x for Point of product G;
reserve X for set;

theorem Th27:
for i be set st i in dom G holds
  r(#)(f*reproj(In(i,dom G),x)) = (r(#)f)*reproj(In(i,dom G),x)
proof
   let i0 be set;
   assume i0 in dom G;
   set i=In(i0,dom G);
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 G.i by FUNCT_2:def 1;
A4b:   for s be Element of G.i holds s in dom((r(#)f)*reproj(i,x))
     iff s in dom(f*reproj(i,x))
   proof
    let s be Element of G.i;
    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
A4:   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
A4a:dom(r(#)(f*reproj(i,x))) =dom((r(#)f)*reproj(i,x)) by TARSKI:2;
A5:for s be Element of G.i holds s in dom((r(#)f)*reproj(i,x))
     iff reproj(i,x).s in dom(r(#)f)
   proof
    let s be Element of G.i;
    dom(reproj(i,x))=the carrier of G.i by FUNCT_2:def 1;
    hence thesis by FUNCT_1:11;
   end;
   for z being Element of G.i 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 G.i;
    assume A6: z in dom(r(#)(f*reproj(i,x))); then
A7: z in dom(f*reproj(i,x)) by VFUNCT_1:def 4;
A9: f/.(reproj(i,x).z) = f.(reproj(i,x).z) by A1,A5,A4a,A6,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 A2,A4b,A6,FUNCT_1:12
      .= (r(#)f)/.(reproj(i,x).z) by A5,A4a,A6,PARTFUN1:def 6
      .= r * f/.(reproj(i,x).z) by A5,A4a,A6,VFUNCT_1:def 4;
    hence thesis by A10;
   end;
   hence thesis by A4,TARSKI:2,PARTFUN1:5;
end;
