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 Th22:
  f=h & x=z implies Proj(j,n)*f*reproj(i,x) = <>*(proj(j,n)*h* reproj(i,z))
proof
  reconsider h1=proj(1,1)qua Function" as Function of REAL,REAL 1 by Th2;
  assume that
A1: f=h and
A2: x=z;
  <>*(proj(j,n)*h*reproj(i,z)) = h1*(proj(j,n)*(h*reproj(i,z)))*proj(1,1)
  by RELAT_1:36
    .= h1*proj(j,n)*(h*reproj(i,z))*proj(1,1) by RELAT_1:36
    .= h1*proj(j,n)*(h*reproj(i,z)*proj(1,1)) by RELAT_1:36
    .= h1*proj(j,n)*(h*(reproj(i,z)*proj(1,1))) by RELAT_1:36
    .= Proj(j,n)*(h*(reproj(i,z)*proj(1,1))) by Th11
    .= Proj(j,n)*(h*reproj(i,x)) by A2,Th12;
  hence thesis by A1,RELAT_1:36;
end;
