reserve S,T,W,Y for RealNormSpace;
reserve f,f1,f2 for PartFunc of S,T;
reserve Z for Subset of S;
reserve i,n for Nat;

theorem LM190:
  for X, Y be RealNormSpace,
  z be Point of [:X,Y:] holds
  z`1 = proj(In(1,dom<*X,Y*>)).(IsoCPNrSP(X,Y).z) &
  z`2 = proj(In(2,dom<*X,Y*>)).(IsoCPNrSP(X,Y).z)
  proof
    let X, Y be RealNormSpace,
    z be Point of [:X,Y:];
    set i1 = In(1,dom <*X,Y*>);
    set i2 = In(2,dom<*X,Y*>);
    D1: dom <*X,Y*> = Seg len <*X,Y*> by FINSEQ_1:def 3
    .= Seg 2 by FINSEQ_1:44;
    then 1 in dom <*X,Y*>;
    then
    AS1: i1 = 1 by SUBSET_1:def 8;
    2 in dom <*X,Y*> by D1;
    then
    AS2:i2 = 2 by SUBSET_1:def 8;
    set G = <*X,Y*>;
    AS3: product G = NORMSTR(# product carr G,zeros G,[:addop G:]
    ,[:multop G:], productnorm G #) by PRVECT_2:6;
    consider x be Point of X, y be Point of Y such that
    P1: z=[x,y] by PRVECT_3:18;
    P4: IsoCPNrSP(X,Y).z = IsoCPNrSP(X,Y).(x,y) by P1
    .=<*x,y*> by defISO;
    reconsider w = IsoCPNrSP(X,Y).z as Element of product carr G by AS3;
    P5: proj(In(1,dom<*X,Y*>)).w = <*x,y*> .1 by P4,AS1,NDIFF_5:def 3
    .= z`1 by P1;
    proj(In(2,dom<*X,Y*>)).w = <*x,y*> .2 by P4,AS2,NDIFF_5:def 3
    .= z`2 by P1;
    hence thesis by P5;
  end;
